The Local Pro-p Anabelian Geometry of Curves.pdf

The Local Pro-p Anabelian Geometry of Curves by Shinichi Mochizuki INTRODUCTION The Anabelian Geometry of Grothendieck: Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π 1 (X). This group π 1 (X) is a proﬁnite group which is uniquely determined (up to inner automorphisms) by the property that the category of ﬁnite, discrete sets equipped with a continuous π 1 (X)-action is equivalent to the category of ﬁnite étale coverings of X. Moreover, the assignment X → π 1 (X) is a functor from the category of connected schemes (and morphisms of schemes) to the category of proﬁnite topological groups and continuous outer homomorphisms (i.e., continuous homomorphisms of topological groups, where we identify any two homomorphisms that can be obtained from one another by composition with an inner automorphism). Now let K be a ﬁeld. Let Γ K be the absolute Galois group of K. Then π 1 (Spec(K)) may be identiﬁed with Γ K . Let X K be a variety (i.e., a geometrically integral separated scheme of ﬁnite type) over K. Then the structure morphism X K → Spec(K) deﬁnes a natural augmentation π 1 (X K ) → Γ K . The kernel of this morphism π 1 (X K ) → Γ K is a closed normal subgroup of π 1 (X K ) – called the geometric fundamental group of X K – def which may be identiﬁed with π 1 (X K ) (where X K = X K ⊗ K K). If, moreover, one ﬁxes a prime number p, then one can form the maximal pro-p quotient π 1 (X K ) (p) of π 1 (X K ). Since the quotient π 1 (X K ) → π 1 (X K ) (p) is characteristic, it follows that the kernel of this quotient is, in fact, a normal subgroup of π 1 (X K ). The quotient of π 1 (X K ) by this normal subgroup will be denoted Π X K . Thus, Π X K inherits a natural augmentation Π X K → Γ K from that of π 1 (X K ). Now let us consider the assignment π 1 (−) K : {X K → Spec(K)} → {π 1 (X K ) → Γ K } This assignment deﬁnes a functor from the category C K of K-varieties (whose morphisms are K-linear morphisms of varieties) to the category G K whose objects are proﬁnite topo- logical groups equipped with an augmentation to Γ K , and whose morphisms are continu- ous outer homomorphisms of topological groups that lie over Γ K . It was the intuition of Grothendieck (see [Groth]) that: 1 For certain types of K, if one replaces C K and G K by “certain appro- and G K (such that π 1 (−) K still maps C K into priate” subcategories C K G K ), then π 1 (−) K should be fully faithful. Here, the “certain appropriate” subcategories C K for which this piece of intuition was to hold true were tentatively assigned the appellation anabelian, while the piece of intuition itself came to be referred to as Grothendieck’s Conjecture of Anabelian Geometry (or, sim- ply, the “Grothendieck Conjecture,” for short). Roughly speaking, the sorts of varieties that were thought to be likely to be “anabelian” were varieties that are “suﬃciently hyper- bolic.” (Note that, as one can see in the case of curves, hyperbolic varieties tend to have highly nonabelian fundamental groups, hence the term “anabelian.” In higher dimensions, however, things are not so simple – see, e.g., [IN].) A variant of the above “proﬁnite” Grothendieck Conjecture is the following “pro- p” Grothendieck Conjecture: Namely, instead of considering π 1 (−) K , one considers the functor that assigns to the K-variety X K the augmented group Π X K → Γ K . The “pro-p” Grothendieck Conjecture then asserts that, in certain situations, this functor should be fully faithful. The present paper is concerned with proving various versions of the proﬁnite and pro-p Grothendieck Conjectures under various conditions. Statement of the Main Results: Let p be a prime number. Let K be a sub-p-adic ﬁeld (cf. Deﬁnition 15.4 (i)), i.e., a subﬁeld of a ﬁnitely generated ﬁeld extension of Q p . In this paper, we prove a pro-p version of the Grothendieck Conjecture (Theorem A) for dominant morphisms between “smooth pro-varieties” and “hyperbolic pro-curves” over K. (Here, by a “smooth pro-variety” (respectively, “hyperbolic pro-curve”) over K, we mean a K-scheme which can be written as a projective limit of smooth varieties (respectively, hyperbolic curves) over K in which the transition morphisms are birational – cf. Deﬁnitions 15.4 (ii) and 16.4.) We then give various versions of Theorem A (namely, Theorems A and A ) for truncated fundamental groups. From Theorem A, we also derive a proﬁnite version of the Grothendieck Conjecture (Theorem B) for morphisms between function ﬁelds (of arbitrary dimension) over K. Next, we apply Theorem A to prove the “injectivity part” (Theorem C) of the pro-p “Section Conjecture.” Finally, in an Appendix, we derive from Theorem A an isomorphism version of the Grothendieck Conjecture for certain hyperbolic surfaces (Theorem D). def Notation: If K is a ﬁeld and X K is a K-scheme, we denote by Π prf X K = π 1 (X K ) the fundamental group of X K (for some choice of base-point), and by Γ K the absolute Galois group of K. Then we have a natural morphism Π prf X K → Γ K whose kernel is the geometric prf prf fundamental group Δ X ⊆ Π X K . Let Δ X be the maximal pro-p quotient of Δ prf X . Then prf prf the kernel of Δ X → Δ X is a normal subgroup of Π X K , so by forming the quotient of Π prf X K by this normal subgroup, we obtain a group Π X K , together with a morphism Π X K → Γ K def whose kernel is Δ X . Next, if Δ is a topological group, we let: Δ{0} = Δ; for i ≥ 1, def Δ{i} = [Δ{i − 1}, Δ{i − 1}] (here we mean “the closed subgroup generated by the purely 2 group-theoretic commutator subgroup”). In the speciﬁc case of fundamental groups, let us write “Π i ” (respectively, “Δ i ”) for Π/Δ{i} (respectively, Δ/Δ{i}). Our ﬁrst main theorem is the following: Theorem A. Let p be a prime number. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let X K be a smooth pro-variety over K. Let Y K be a hyperbolic pro-curve over K. Let Hom dom K (X K , Y K ) be the set of dominant K-morphisms from X K to Y K . Let open prf prf Hom Γ K (Π X K , Π Y K ) (respectively, Hom open Γ K (Π X K , Π Y K )), be the set of open, continuous prf group homomorphisms Π X K → Π Y K (respectively, Π prf X K → Π Y K ) over Γ K , considered up to composition with an inner automorphism arising from Δ Y (respectively, Δ prf Y ). Then the natural maps open prf prf Hom dom K (X K , Y K ) → Hom Γ K (Π X K , Π Y K ) → Hom open Γ K (Π X K , Π Y K ) are bijective. This Theorem is given as Theorem 16.5 (cf. also the Remark following Theorem 16.5) in the text. It is from Theorem A that all of the other major results of this paper are derived. We remark that: (1) In fact, really, the main portion of Theorem A is the bijectivity of the ﬁrst and third Hom’s. That is to say, the bijectivity of the ﬁrst and second Hom’s follows formally from the bijectivity of the ﬁrst and third Hom’s. See the Remark following Theorem 16.5 for more details. (2) The notion of a “smooth pro-variety” (respectively, “hyperbolic pro- curve”) has as special cases: (i) a smooth variety (respectively, hyper- bolic curve) over K; (ii) the spectrum of a function ﬁeld of arbitrary dimension (respectively, function ﬁeld of dimension one) over K. (3) There exists a substantial body of people who, when they speak of “the Grothendieck Conjecture” (respectively, “the pro-p Grothendieck Conjecture”), refer to the following rather speciﬁc statement (which is a special case of the general philosophy discussed in the preceding subsection): If K is a ﬁnitely generated extension of Q, and X K and Y K are either hyperbolic curves over K or the spectra of one-dimensional function ﬁelds over K, then the isomorphisms of X K with Y K are in natural bijective correspondence with the outer isomorphisms over Γ K of prf Π prf X K (respectively, Π X K ) with Π Y K (respectively, Π Y K ). This statement is manifestly a special case of the proﬁnite version – see (1) above – (respectively, pro-p version, i.e., version as stated above) of Theorem A. 3 We also have truncated versions of Theorem A (Theorems 18.1 and 18.2 in the text): Theorem A . Let K be sub-p-adic. Let X K be a smooth variety over K. Let Y K be a hyperbolic curve over K. Let n ≥ 5. Then every continuous open homomorphism θ : Π nX K → Π nY K over Γ K induces a dominant morphism μ : X K → Y K whose induced morphism on fun- damental groups coincides (up to composition with an inner automorphism arising from n−3 Δ Y K ) with the morphism Δ n−3 X K → Δ Y K deﬁned by considering θ “modulo Δ{n − 3}.” Let K be sub-p-adic. Let X K be a smooth pro-variety over K. Let Y K Theorem A . be a hyperbolic pro-curve over K. Let n 0 be the minimum transcendence degree over Q p of all ﬁnitely generated ﬁeld extensions of Q p that contain K. Let n 0 be the transcendence def degree over K of the function ﬁeld of X K . Let n 0 = n 0 + 2(n 0 − 1) + 1. Let n ≥ 3n 0 + 5. Then every continuous open homomorphism θ : Π nX K → Π nY K over Γ K induces a dominant morphism μ : X K → Y K whose induced morphism on fundamental groups coincides (up to composition with an inner automorphism arising 0 0 → Δ n−3−3n deﬁned by considering θ “mod- from Δ Y K ) with the morphism Δ n−3−3n X K Y K ulo Δ{n − 3 − 3n 0 }.” Our second main theorem (Theorem 17.1 in the text) is the following: Theorem B. Let p be a prime number. Let K be sub-p-adic. Let L and M be function ﬁelds of arbitrary dimension over K. (In particular, we assume that K is alge- braically closed in L and M.) Let Hom K (Spec(L), Spec(M)) be the set of K-morphisms from M to L. Let Hom open Γ K (Γ L , Γ M ) be the set of open, continuous group homomorphisms Γ L → Γ M over Γ K , considered up to composition with an inner automorphism arising from Ker(Γ M → Γ K ). Then the natural map Hom K (Spec(L), Spec(M)) → Hom open Γ K (Γ L , Γ M ) is bijective. Note that in characteristic zero, this generalizes the results of [Pop1], [Pop2], where a similar result to Theorem B is obtained, except that the morphisms Γ L → Γ M are required to be isomorphisms, and K is required to be ﬁnitely generated over Q. 4 Our third main theorem (Theorem 19.1 in the text) is the following: Theorem C. Let p be a prime number. Let K be sub-p-adic. Let X K be a hyperbolic curve over K. Let X K (K) be the set of K-valued points of X K . Let Sect(Γ K , Π X K ) be the set of sections Γ K → Π X K of Π X K → Γ K , considered up to composition with an inner automorphism arising from Δ X . Then the natural map X K (K) → Sect(Γ K , Π X K ) is injective. Finally, in the Appendix, we use Theorem A to derive the following result: Theorem D. Let p be a prime number. Let K be sub-p-adic. Let X K and Y K be hyperbolically ﬁbred surfaces (see Deﬁnition a2.1 in the Appendix for a precise deﬁnition of this term) over K. Let Isom K (X K , Y K ) be the set of K-isomorphisms (in the category prf of K-schemes) between X K and Y K . Let Isom Γ K (Π prf X K , Π Y K ) be the set of continuous prf group isomorphisms Π prf X K → Π Y K over Γ K , considered up to composition with an inner automorphism arising from Δ prf Y . Then the natural map prf Isom K (X K , Y K ) → Isom Γ K (Π prf X K , Π Y K ) is bijective. Recent Work on the Grothendieck Conjecture: In this subsection, we would like to take a brief look at recent work on the Grothendieck Conjecture. We will concentrate only on major and recent (since the early 1980’s) develop- ments that relate to the present paper, and we make no pretense of giving a complete history of work on the Grothendieck Conjecture. Before beginning, it is worth pointing out that until the appearance of the present paper (and its earlier version [Mzk2]), it was widely assumed that the base ﬁeld “K” that appears in the Grothendieck Conjecture should be assumed to be ﬁnitely generated over Q. It is not clear precisely why this came to be as- sumed by most people working in the ﬁeld, but one possible cause is that the Grothendieck Conjecture appears to have originated as an alternative approach to diophantine geometry ([Groth], [Pop3]). Thus, if one’s ultimate aim is applications to diophantine geometry, it is quite natural only to look at global ﬁelds (i.e., ﬁnitely generated extensions of Q), and not at p-adic local ﬁelds, for instance. Another reason for the ﬁxation on ﬁnitely generated extensions of Q appears to have been that many people conceived of the Grothendieck Conjecture as an anabelian version 5 of the Tate Conjecture (Faltings’ theorem – see [Falt3]) for abelian varieties over number ﬁelds. In fact, this point of view became so deeply engrained that it gave rise to a tendency for people to try to prove the Grothendieck Conjecture for hyperbolic curves over number ﬁelds by deducing it from the Tate Conjecture for abelian varieties over number ﬁelds (see, e.g., [Naka3]). In fact, even important early work – such as [Naka1] on the Grothendieck Conjecture for genus zero hyperbolic curves over number ﬁelds, or [Pop1], [Pop2] which treat the birational case – which does not try to deduce the Grothendieck Conjecture from the Tate Conjecture still had a distinctly global ﬂavor, and relied extensively on essentially global techniques. These global techniques ultimately proved to be rather irrelevant to the proof of the main results of the present paper. Nevertheless, what was important (from the point of view of the author) about the work of H. Nakamura was that it established a host of basic techniques for studying the outer Galois action on the (nonabelian) geometric fundamental group of a hyperbolic curve, which was unknown territory to most arithmetic geometers, who were only familiar with the Galois action on the abelian fundamental group of an abelian variety. It was this culture of basic techniques (due to H. Nakamura) which permitted the subsequent development of more powerful approaches to the Grothendieck Conjecture itself, as discussed below. The next important development (in early 1995) was the work of A. Tamagawa ([Tama]) in which it was shown that the isomorphism class of any aﬃne hyperbolic curve over an absolutely ﬁnitely generated ﬁeld is functorially determined by the outer Galois ac- tion on its proﬁnite geometric fundamental group. (In fact, by a relatively straightforward argument ([Mzk1]), it is possible to remove the “aﬃneness” hypothesis in Tamagawa’s result, at least in characteristic zero.) Unlike the results of the present paper, Tamagawa’s result holds in positive characteristic as well as in characteristic zero. Moreover, Tama- gawa’s ideas on characterizing those sections of π 1 (X K ) → Γ K (where X K is a hyperbolic curve over a ﬁeld K) that arise from geometric points of X K played a pivotal role in inspir- ing the author to prove the results of the present paper. Concretely speaking, the inﬂuence of these ideas of Tamagawa can be seen in the argument of Section 9 of the present paper. Finally, we turn to discussing the results of the present paper in a historical context. As observed previously, the results of the present paper are (unlike Tamagawa’s results) only valid in characteristic zero. On the other hand, the main advances of Theorem A of the present paper relative to [Tama] are as follows: (1) The hyperbolic curves involved are allowed to be proper. In fact, we even allow what we call hyperbolic pro-curves. (2) Instead of dealing with proﬁnite geometric fundamental groups, we deal with pro-p geometric fundamental groups. Pro-p results tend to be stronger than proﬁnite results in the sense that proﬁnite results usu- ally follow immediately from pro-p results (see the Remark following Theorem 16.5 for more details). (3) Instead of just considering isomorphisms, we allow arbitrary dominant morphisms between the varieties involved. 6 (4) The variety “on the left” (in the “Hom(−, −)” of Theorem A) is allowed to be higher-dimensional, and need not even be hyperbolic. (5) The base ﬁeld can be any subﬁeld of a ﬁnitely generated extension of Q p , whereas for Tamagawa (in characteristic zero) the base ﬁeld must be a ﬁnitely generated extension of Q. From the point of view of the author, (5) is the most fundamental and important advance as it is associated with the fact that the techniques of the present paper are fundamentally diﬀerent from those of both Nakamura and Tamagawa in that they are couched in the world of p-adic Hodge theory. That is to say, the proof of Theorem A may be regarded as an application of the theory of [Falt1] and [BK]. It is precisely the use of these p-adic techniques that allowed the author to prove a result which was much stronger (in the above ﬁve senses) than the result of [Tama] (in characteristic zero) or [Mzk1]. Moreover, it is the opinion of the author that: The reason that it took so long for Theorem A to be discovered was the overwhelming prejudice of most people in the ﬁeld that the Grothendieck Conjecture for hyperbolic curves is an essentially global result, akin to the Tate Conjecture for abelian varieties over number ﬁelds. In fact, however, it is much more natural to regard the Grothendieck Conjecture for hyperbolic curves as an essentially local, p-adic result that belongs to that branch of arithmetic geometry known as p-adic Hodge theory. Moreover, it is the feeling of the author that, more than the technical details of the state- ment of Theorem A, it is this fact – i.e., that the Grothendieck Conjecture for hyperbolic curves is best understood not as a global, number-theoretic result, but rather as a result in p-adic Hodge theory – that is the central discovery of this paper. The Structure of the Proof: First, we remark that most of the paper (Sections 1 through 14) is devoted to proving the following technical result: (∗) tech Suppose that K is a ﬁnite extension of Q p . Let X K and Y K be proper hyperbolic curves over K. Let U K be the spectrum of the function ﬁeld of X K . Then every continuous surjective homomorphism θ : Π U K → Π Y K over Γ K arises from some geometric morphism U K → Y K . Section 15 is devoted to the rather standard technicalities necessary to generalize (∗) tech to the case where K is any subﬁeld of a ﬁnitely generated extension of Q p . Section 16 is devoted to carrying out a standard “cutting by hyperplane sections argument” which allows one to replace U K by a higher-dimensional smooth pro-variety (which thus completes 7 the proof of Theorem A). Section 17 is devoted to an induction argument used to derive Theorem B from Theorem A. Section 18 discusses how the techniques used to prove Theorem A in fact also give rise to the truncated versions Theorem A and Theorem A . Finally, Section 19 reviews a standard argument (already present in the work of H. Nakamura) which allows one to derive Theorem C from Theorem A. Thus, the rest of this subsection will be devoted to outlining the proof of (∗) tech in Sections 1 through 14. In order to simplify the discussion, let us consider ﬁrst the following slightly modiﬁed version of (∗) tech : (∗) prop Suppose that K is a ﬁnite extension of Q p . Let X K and Y K be proper hyperbolic curves over K. Then every continuous surjective homomorphism θ : Π X K → Π Y K over Γ K arises from some geometric morphism X K → Y K . (Here, “prop” stands for “proper” – i.e., since, unlike in (∗) tech , where U K appeared, in (∗) prop only proper curves appear.) Thus, (∗) tech implies (∗) prop . In other words, (up to some standard general nonsense) one may think of (∗) tech as the concatenation of (∗) prop with the following assertion: (∗) iner Any θ : Π U K → Π Y K as in (∗) tech necessarily factors through the quotient Π U K → Π X K . (Here, “iner” stands for “inertia” – since the kernel of Π U K → Π X K is generated by inertia groups, so (∗) iner is the assertion that θ : Π U K → Π Y K always sends inertia groups in Π U K to the identity.) We will come back to the issue of (∗) iner later, but for now, let us concentrate on outlining the proof of (∗) prop , since it is this which is the core of the paper. Thus, let us assume that we have been given a continuous surjective homomorphism: θ : Π X K → Π Y K over Γ K . We would like to somehow manufacture out of θ a morphism from X K to Y K . A natural, naive way to start is the following: First, by replacing X K and Y K by appropriate étale coverings, we may assume that neither is hyperelliptic (see, e.g., Lemma 10.4 (4)). Let us write D X (respectively, D Y ) for H 0 (X K , ω X K /K ) (respectively, H 0 (Y K , ω Y K /K )), the space of global diﬀerentials on X K (respectively, Y K ). Then it is well-known that X K def (respectively, Y K ) embeds naturally in the projective space P X = P(D X ) (respectively, def P Y = P(D Y )). Moreover, since the p-adic étale cohomology of X K or Y K can be computed as the group cohomology of the respective pro-p geometric fundamental group, it follows that the surjection θ induces a Γ K -equivariant injection θ H : H 1 (Y K , Q p ) → H 1 (X K , Q p ) 8 Moreover, it is well-known (see, e.g., [Falt1], [Tate]) that the Γ K -module H 1 (X K , Q p ) is Hodge-Tate, and, moreover, that if we tensor H 1 (X K , Q p ) over Q p with C p (1) (where the “(1)” is a “Tate twist”), and take Γ K -invariants, we naturally recover the space D X of diﬀerentials. Thus, since θ H is Γ K -equivariant, if we tensor θ H over Q p with C p (1) and take Γ K -invariants, we obtain from θ (in a natural way) a K-linear injection θ D : D Y → D X and hence a rational map from P X to P Y . Thus, in some sense, without doing anything terribly new (i.e., we have only just applied results known to Tate since the 1960’s), we have already come relatively close to constructing a morphism from X K to Y K . Indeed, what we have done is to construct a morphism θ D from diﬀerentials on Y K to diﬀerentials on X K , which we would like to hope arises as the pull-back map on diﬀerentials associated to a morphism from X K to Y K . Moreover, since X K (respectively, Y K ) is canonically embedded in P X (respectively, P Y ), it follows from elementary algebraic geometry that the “only” thing we need to show is ⊗i i def that θ D preserves relations: That is, if i is a positive integer, let D X = H 0 (X K , ω X ). K /K i Similarly, we have D Y . Note that multiplication of diﬀerential forms deﬁnes a natural morphism i D Y → D Y i Let us denote the kernel of this morphism by R i . Thus, R i is the set of relations (of degree i) deﬁning Y K as a subvariety of P Y . Moreover, by composing the i th tensor power of θ D i with the natural morphism from the i th tensor power of D X to D X , we obtain a morphism i κ : i i D Y → D X Then we shall say that θ : Π X K → Π Y K preserves relations if κ i (R i ) = 0 for all positive integers i. As stated earlier, once we know that θ preserves relations, it follows from elementary algebraic geometry (see [Harts], Chapter II) that the rational map from P X to P Y deﬁned by θ D induces a (dominant) morphism X K → Y K , as desired. (Once one has this morphism X K → Y K , the fact that the map that it induces on fundamental groups coincides with θ is a matter of general nonsense – for details, we refer to the argument preceding Theorem 14.1.) Thus, to review what we have done so far, we have reduced the main problem to showing that any θ : Π X K → Π Y K as in (∗) prop preserves relations. It should be emphasized at this point, that so far everything that we have done has been painless general nonsense – the substantive mathematical core of the argument is yet to come (after some more general nonsense in the next few paragraphs). The next step is to introduce a ﬁeld L, 9 as follows: Let us assume for simplicity that X K extends to a stable curve X over O K . Let p ∈ X be an irreducible component of the special ﬁber of X . Then the completion of the local ring O X , p is a p-adically complete discrete valuation ring O L , whose quotient ﬁeld we denote by L. The space Ω L of p-adically continuous diﬀerentials of L over K is then a one-dimensional L-vector space. Thus, L is, in some sense, one-dimensional over K. Alternatively, one may think of Spec(L) roughly as some sort of small p-adic open set in X K . At any rate, from the construction of L, it follows that we have a natural L-valued point ξ X : Spec(L) → X K . Since L is “one-dimensional,” it is easy to check that the restriction map on diﬀerentials i D X → Ω ⊗i L (where “Ω ⊗i L ” denotes the tensor product of i copies of Ω L over L) is injective. Thus, instead of checking that κ i (R i ) = 0, it suﬃces to check that the composite κ iL : i D Y → Ω ⊗i L of κ i with the above restriction map vanishes on R i . In other words, we would like to compute κ iL . To “compute” κ iL , we need to go back to looking at fundamental groups. First of all, let us observe that by functoriality of the fundamental group, the natural L-valued point ξ X : Spec(L) → X K deﬁnes a morphism α L X : Γ L → Π X K whose composite with the augmentation Π X K → Γ K is the morphism Γ L → Γ K on Galois groups induced by the inclusion of ﬁelds K ⊆ L. Moreover, if we compose α L X with θ, we obtain a morphism α L Y : Γ L → Π Y K Now let us suppose that we know that α L Y is geometric, i.e., that it arises from some L- valued point ξ Y : Spec(L) → Y K of Y K . Then it follows immediately from the theory of [Falt1] that κ iL may be computed as the restriction map on diﬀerentials associated to ξ Y . But it is clear that this restriction map on diﬀerentials annihilates R i . Thus, to summarize: in order to show that θ preserves relations, it suﬃces to show that α L Y is geometric. As remarked earlier, the above prefatory remarks are just “general nonsense.” The mathematical core of the present paper lies in showing that given a geometric α L X , together prop L , the resulting α Y is again geometric. The proof that with a θ : Π X K → Π Y K as in (∗) L α Y is geometric is long and intricate, and can be divided roughly into four parts: 10 (1) First, we consider a K-valued point x ∈ X K (K) of X K , and its asso- ciated map on fundamental groups α K X : Γ K → Π X K . It is elementary to show that the process of passing from the point x ∈ X K (K) to the arithmetic ﬁrst Chern class of its associated line bundle def K 2 = c 1 (O X K (x)) ∈ H et (X K , Z p (1)) = H 2 (Π X K , Z p (1)) η X can be carried out just by working with α K X (see the ﬁrst half of Section 7 K for details on this “group-theoretic recipe” for concocting η X out of α K X ). K K On the other hand, by composing α X with θ, we get an α Y : Γ K → Π Y K . The ﬁrst step then is to show that: If one carries out this recipe for α K Y (which may or may not arise geometrically) so as to obtain a class η Y K ∈ H 2 (Π Y K , Z p (1)), then η Y K is the arithmetic ﬁrst Chern class of a line bundle (necessarily of degree prime to p) on Y K . This is the topic of Sections 1 through 5. L (2) Next, we go back to L-valued points, and their associated α L X , α Y . L Then, just as in the case of K-valued points, one can form η Y , and one would like to know that η Y L is the arithmetic ﬁrst Chern class of def a line bundle (necessarily of degree prime to p) on Y L = Y K ⊗ K L. This is technically much more intricate than the K-valued case, and is done (roughly speaking) by thinking about the “diﬀerence” between the Chern class of a K-valued point and of an L-valued point, and showing that this diﬀerence has special properties that are preserved by θ. This allows one to derive the assertion in the L-valued case from the corresponding assertion in the K-valued case, which was already handled in (1) above. This reduction is the topic of Sections 6 and 7. (3) Recall that we would like to show that α L Y is geometric. We know from (2) above that at least there exists a line bundle on Y L of degree prime to p. Then an elementary algebraic geometry argument shows that this implies the existence of a rational point of Y L deﬁned over a tamely ramiﬁed (this will be crucial in Step (4)!) extension L of L. Although this portion of the proof is technically rather trivial, its discovery was a key step in the creation of the proof of Theorem A. This portion of the proof is discussed in Section 8. (4) Finally, by applying (3) to all the curves in a certain tower of coverings of the original Y L , we obtain a collection of rational points of the curves of this tower that are deﬁned over tamely ramiﬁed extensions of L. Then by using Faltings’ p-adic Hodge theory ([Falt1]), we show that these rational points necessarily converge p-adically to a single L-valued point of Y L = Y K ⊗ K L whose associated Γ L → Π Y K is necessarily equal to 11 L α L Y . This completes the proof of the geometricity of α Y . This portion of the proof is given in Sections 9 and 10 (and applied in Section 13). As discussed above, once one knows the geometricity of α L Y , one can conclude the preser- vation of relations – this is discussed in Sections 11 through 13. Finally, the proof of (∗) iner (i.e., the diﬀerence between (∗) tech and (∗) prop ) is given in Section 14. Unfortunately, in order to show (∗) iner , it is necessary to go through all the steps discussed so far for a given θ : Π U K → Π Y K , and then to conclude (∗) iner from the preservation of relations. This makes the proof much more technically intricate than it would be if one could prove (∗) iner from some sort of a priori argument, and then prove preservation of relations only for θ : Π X K → Π Y K as in (∗) prop . We would like to close this outline of the proof of Theorem A by discussing Step (1) above (i.e., the content of Sections 1 through 5) in greater detail. The reason for this is that Step (1) is what allowed us to generalize the “isomorphisms only” result of [Mzk2] to the homomorphism result of the present paper. First of all, let us recall the notion of the Malčev completion (cf., e.g., [Del], §9) of Δ X . In fact, we shall only need the “smallest nontrivial part of Malčev completion”: concretely, a unipotent algebraic group over Q p , which we denote by M X , whose representations are the same as continuous representations of Δ X /[Δ X , [Δ X , Δ X ]] on some Q p -vector space V which is equipped with a Δ X -invariant ﬁltration on whose subquotients Δ X acts trivially. In fact, since unipotent algebraic groups are equivalent to their Lie algebras, we shall consider instead the Lie algebra M X of M X . Next, observe that any section α K X : Γ K → Π X K deﬁnes (by conjugation) a true action (i.e., not just an action up to inner automorphisms) of Γ K on Δ X , hence on M X . Relative to this action, their exists a unique “weight zero quotient” M X ⊗ Q p C p → Z X . Here, we call the quotient “weight zero” because it is the maximal quotient M X ⊗ Q p C p → Q for which the action of Γ K on Q is such that Q has a ﬁltration by Γ K -submodules whose subquotients are Γ K -equivariantly isomorphic to C p . Moreover, this quotient M X ⊗ Q p C p → Z X is independent of the choice of section α K X . Step (1) is based on the following pair of observations: (i) On the one hand, if α K X arises geometrically, then Z X is “Hodge- Tate.” (By abuse of terminology, we use the term “Hodge-Tate” here to mean that Z X has a C p -basis of Γ K -invariant elements.) This is the content of Proposition 3.5. K is the (ii) On the other hand, if Z X is “Hodge-Tate,” then the class η X ﬁrst Chern class of a line bundle. This is essentially the content of the calculation performed in Proposition 4.4 (see also Lemma 7.3). Moreover, if one starts with a surjective θ : Π X K → Π Y K , then Z X maps naturally to, and in fact, surjects onto Z Y . Thus, if one starts with an α K X that arises geometrically, then one knows from Observation (i) above that Z X is Hodge-Tate, but any quotient of a Hodge-Tate representation of Γ K – i.e., such as Z Y – is always Hodge-Tate, so Observation (ii) above thus allows one to complete Step (1). 12 At this point, the reader may wonder how the author stumbled upon the two key observations of the preceding paragraph. In fact, the author ﬁrst realized what was going on by considering the ordinary case (see Section 1 for a detailed discussion). In this case, the “weight zero quotient” exists at the level of groups, without passing to Malčev completions or tensoring with C p : namely, (in the notation of Section 1) it is the quotient et Δ X → Δ et X (which thus gives rise to a quotient Π X K → Π X K ). Moreover, as is shown in Section 1, it is elementary to show that Δ X → Δ et X can be recovered group-theoretically, and that every section Γ K → Π X K that arises geometrically (from a point of X K ) induces et a ﬁxed, group-theoretically constructible section θ X : Γ K → Π et X K of Π X K → Γ K . These observations are enough to complete Step (1) (in the ordinary case). Thus, one may regard the discussion in the preceding paragraphs (which is valid in the nonordinary case) as simply the result of generalizing the observations discussed in this paragraph in the ordinary case to the possibly nonordinary case by means of the technical machinery of the Malčev completion and p-adic Hodge theory. Finally, let us make the following observation: As one can see from the key argument discussed above, in fact, really, one does not need all of Δ X . That is to say, in the above argument (concerning the issue of when Z X is Hodge-Tate), one actually only uses the quotient Δ X /[Δ X , [Δ X , Δ X ]]. It is this observation that is behind the truncated generalizations (Theorems A and A ) of Theorem A. Acknowledgements: I would like to thank A. Tamagawa: (i) for inspiring me by means of his paper [Tama]; (ii) for numerous discussions during which I presented the proof of the main results of the present paper; (iii) for advice on surmounting two technical problems (see the Remark at the end of Section 12; Lemmas 15.6 and 15.8); (iv) for explaining to me the “general nonsense argument” preceding Theorem 14.1; (v) for explaining to me basic facts concerning the Malčev completion, which were of immense importance in developing the theory discussed in this paper. Also, I would like to thank T. Saito and G. Faltings for pointing out various minor errors in earlier versions of this manuscript. Section 0: Preliminaries and Notations Let p be a prime number. Throughout this paper the symbol “∧” over or to the upper- right of an object will denote the p-adic completion of that object. Let K be a Q p -algebra. Let Ω be an algebraically closed ﬁeld. Then given a base-point b ∈ Hom Ring (K, Ω), we can form the algebraic fundamental group π 1 (Spec(K), b) Typically, the choice of base-point b will not be important for us, so we shall write Γ K for π 1 (Spec(K), b). Suppose that Spec(K) is a “universal covering space” for Spec(K) such 13 that b arises from a ring homomorphism K → Ω. Then we may think of Γ K as Gal(K/K). We shall denote the étale cohomology of Spec(K) by H i (K, −). If M is a proﬁnite abelian group with a continuous Γ K -action, then M naturally deﬁnes an inverse system {F α } of locally constant sheaves on the étale site of Spec(K), and we shall write H i (K, M) for the inverse limit (over α) of the H i (K, F α ). Note that for each r ∈ Z, Z p (r) (where the “(r)” is a Tate twist) has a natural structure of Γ K -module. Deﬁnition 0.1. We shall call K a p-adic ﬁeld if it is the quotient ﬁeld of a p-adically complete, mixed characteristic discrete valuation ring O K . We shall denote the residue ﬁeld (respectively, maximal ideal) of O K by k (respectively, m K ). We shall call K a p-adic local ﬁeld if k is a ﬁnite ﬁeld. def If K is a p-adic ﬁeld, and K is an algebraic closure of K, then K → Ω = K determines a base-point “b,” and we have Γ K = Gal(K/K). In this case, H i (K, −) is equal to the continuous group cohomology of the proﬁnite group Γ K . If K is a p-adic local ﬁeld, then K is, in fact, a ﬁnite extension of Q p . Now (without any assumptions on the Q p -algebra K), let us assume that we are given a hyperbolic curve X K → Spec(K) over K of type (g, r). (By “curve,” we shall always mean a smooth, one-dimensional, geometrically connected scheme over the base. By “type (g, r),” we mean that X K ⊗ K K is obtained by removing r mutually nonintersecting K- valued points from a proper curve over K of genus g. By “hyperbolic,” we mean that 2g − 2 + r ≥ 1.) When X K is proper, we shall denote its Jacobian (an abelian scheme over K) by J X (or J X K when several bases are in use and it is necessary to specify the base in question). Let us assume that X K is equipped with a base-point x ∈ X K (Ω) (which is compatible def def with the base-point b of Spec(K)). Then we can form Π prf = π 1 (X K , x) and Δ prf X X = π 1 (X K , x). (The use of “Δ” to denote the geometric fundamental group may be new to some readers. Here, we use “Δ” partly because “Π” is already used for the arithmetic fundamental group and partly to conform to the notations of [Falt1], a reference on which the present paper depends heavily.) Occasionally, to avoid confusion, we shall also use the prf prf notation Δ prf X K for Δ X . Let Δ X be the maximal pro-p quotient of Δ X . Since the kernel prf prf of Δ prf X → Δ X is normal in Π X K , we may form the quotient of Π X K by this kernel, and call the resulting quotient group Π X K . Thus, we have an exact sequence 1 → Δ X → Π X K → Γ K → 1 Moreover, this exact sequence induces a representation 14 def ρ X : Γ K → Out(Δ X ) = Aut(Δ X )/Inn(Δ X ) into the outer automorphism group of Δ X . (Here, “Aut(Δ X )” (respectively, “Inn(Δ X )”) denotes the group of continuous automorphisms (respectively, inner automorphisms) of Δ X .) Conversely, it is well-known (see, e.g., [Tama], §7, A.) that the above exact sequence can be recovered from ρ X . Next, we would like to introduce some terminology particular to pro-p groups. Let Δ be a pro-p group (i.e., a topological group obtained by taking an inverse limit of ﬁnite groups of p-power order). Let Δ ⊆ Δ be the unique normal subgroup of Δ with the following property: Δ → Δ/Δ is the maximal (topologically) Hausdorﬀ abelian quotient of Δ which def def is annihilated by p. For i ≥ 0, let Δ <0> = Δ; Δ <i+1> = (Δ ) . Thus, we obtain a descending series of closed normal (even characteristic!) subgroups . . . ⊆ Δ ⊆ . . . ⊆ Δ. Note that since Δ, being a pro-p group, is “pro-solvable,” it follows that the intersection of all the Δ is {1}. Moreover, if Δ is topologically ﬁnitely generated, it follows that the Δ/Δ are all ﬁnite groups. Deﬁnition 0.2. We shall refer to any one of the Δ as a p-derivate of Δ. Thus, in particular, if Δ = Δ X , then it follows (by the structure of the fundamental group of an algebraic curve in characteristic zero) that Δ is topologically ﬁnitely generated, so the Δ/Δ are all ﬁnite groups. Next, let us consider the Kummer sequence on X K , i.e., the exact sequence of étale sheaves on X K given by 0 → Z/p n Z(1) → G m → G m → 0 (for n ≥ 1). (Here, the “(1)” is a “Tate twist,” and the morphism from G m to G m is given by raising to the (p n ) th power.) The connecting morphism induced on étale cohomology by the Kummer sequence then gives us a morphism H 1 (X K , G m ) → H 2 (X K , (Z/p n Z)(1)). Now suppose that L is a line bundle on X K . Applying the connecting morphism just considered to L (which deﬁnes an element of H 1 (X K , G m )), we obtain a compatible system of classes in H 2 (X K , (Z/p n Z)(1)) (for each n ≥ 1), hence a class c 1 (L) ∈ H 2 (X K , Z p (1)). Deﬁnition 0.3. We shall refer to c 1 (L) as the arithmetic ﬁrst Chern class of L. Finally, we have the following elementary technical result, which states that the étale cohomology of a hyperbolic curve may be computed as the group cohomology of its fun- damental group: Lemma 0.4. Assume that K is a ﬁeld. For all integers i, r, the natural morphisms H i (Δ X , Z p (r)) → H i (X K , Z p (r)); H i (Δ X K × K X K , Z p (r)) → H i (X K × K X K , Z p (r)) 15 and H i (Π X K , Z p (r)) → H i (X K , Z p (r)); H i (Π X K × K X K , Z p (r)) → H i (X K × K X K , Z p (r)) are isomorphisms. Proof. By the Leray-Serre spectral sequence, it suﬃces to prove that the morphisms of the ﬁrst line are isomorphisms. Let Z be X K or X K × K X K . Then it follows by general nonsense that it suﬃces to check that for any ﬁnite étale Galois covering Y → Z of p-power order, and any cohomology class η ∈ H i (Y, F p ) (where i > 0), there exists a ﬁnite étale covering Y → Y of p-power order such that η| Y = 0. By the Künneth formula, it suﬃces to do the case Z = X K . Then what we must check is trivial for i > 2 (since then η = 0 to begin with), and clear for i = 1 (by the relationship between étale coverings and H 1 ). If i = 2, then it suﬃces to take Y → Y such that Y → Y has degree p over every connected component of Y . This completes the proof. Section 1: The Ordinary Case Let p be a prime number. Let K be a p-adic ﬁeld with algebraically closed residue ﬁeld. Then, as discussed in Section 0, the absolute Galois group of K will be denoted Γ K . Let X K → Spec(K) be a hyperbolic curve over K of type (g, r). In this Section, let us also assume that X K admits a stable extension X → Spec(O K ) over O K . By this, we mean that there exists a (necessarily unique) r-pointed stable (in particular, proper) curve X → Spec(O K ) of genus g such that X is the complement in X of the images of the r marking sections of X. Let us write X k (respectively, X k ) for X ⊗ O K k (respectively, X ⊗ O K k). Remark. Recall that an r-pointed stable curve of genus g (where 2g − 2 + r ≥ 1) is a proper, ﬂat morphism f : C → S, together with r mutually disjoint sections σ 1 , . . . , σ r : S → C, such that (1) The geometric ﬁbers of f are connected, reduced, of arithmetic genus g, and have at most nodes as singularities. (2) The sheaf ω C/S (σ 1 + . . . + σ r ) (i.e., the sheaf of sections of the dualizing bundle of C over S with poles of order ≤ 1 at the divisors deﬁned by the images of the sections σ 1 , . . . , σ r ) is relatively ample over S. We refer to [DM], [Knud] for more details. 16 Now we make the following Deﬁnition 1.1. We shall call X K ordinary if the Jacobian of every connected component of the normalization of the curve X k is an ordinary abelian variety. In this Section, we would like to assume that X K is ordinary. Under this assumption, it is well-known that π 1 (X k ) (p) (where the “(p)” denotes the maximal pro-p quotient) is a free 2 pro-p group of rank g. (Indeed, this follows from the fact that H et (X k , F p ) = 0 (which may be shown by using the long exact sequence in étale cohomology obtained by considering F p as the kernel of “1− Frobenius” acting on O X k ), plus Proposition 2.3 of Chapter III, §3, of [Sha].) Moreover, since étale coverings of X k lift uniquely to characteristic zero, it follows that we have a continuous surjection: (p) X : Π X K → π 1 (X k ) If we restrict X to Δ X , we obtain a surjection Δ X → π 1 (X k ) (p) . In the following, we shall regard π 1 (X k ) (p) as a quotient of Δ X via this surjection. Let us denote this quotient et by Δ et X . Note that the kernel of the surjection Δ X → Δ X is normal as a subgroup of Π X K . Thus, by taking the quotient of Π X K by this kernel, we obtain a quotient Π X K → Π et X K . In other words, we have an exact sequence et 1 → Δ et X → Π X K → Γ K → 1 together with a surjection et ζ X : Π et X K → Δ X which is the identity on Δ et X . Now observe that the kernel of ζ X projects isomorphically to Γ K . Thus, we obtain a section θ X : Γ K → Π et X K Next, let us observe that every element of Im(θ X ) commutes with every element of Δ et X . Indeed, this follows from the fact that such commutation relations hold after projection et by Π et X K → Γ K , plus the fact that Im(θ X ) = Ker(ζ X ) is normal in Π X K (and maps isomorphically to Γ K via the projection Π et X K → Γ K ). On the other hand, since, as is well-known (see, e.g., [Tama], §1, Propositions 1.1, 1.11), pro-p free groups (of rank ≥ 2) have trivial centers, it thus follows that we obtain the following “group-theoretic” characterization of θ X (when g ≥ 2): 17 et Lemma 1.2. Suppose that g ≥ 2. Then the section θ X : Γ K → Π et X K of Π X K → Γ K is et the unique section whose image commutes with every element of Δ et X ⊆ Π X K . On the other hand, the quotient Δ X → Δ et X (and hence also the quotient Π X K → et Π X K ) can also be reconstructed group-theoretically by means of the following condition on an open normal subgroup H ⊆ Δ X : (∗) et Let N ⊆ Δ X be any subgroup such that H ⊆ N and N/H is cyclic. def Then there exists a surjection N ab = N/[N, N ] → Q, where Q is a free Z p -module of rank one, with the following properties: (i) there exists an open subgroup Γ ⊆ Γ K that stabilizes N and N ab → Q and acts trivially on Q; (ii) the surjection N → N/H factors through Q. Then we have the following “group-theoretic” characterization of the quotient Δ X → Δ et X : Lemma 1.3. The kernel of Δ X → Δ et X is the intersection of all open normal subgroups H ⊆ Δ X such that H satisﬁes the condition (∗) et . Proof. The proof is entirely the same as that of Sections 3 and 8 of [Mzk1]. The basic idea is that if the covering corresponding to Δ X /H is not étale (over O K ), then it has nontrivial inertia subgroups. Such inertia subgroups have nontrivial cyclic subgroups. If we then apply (∗) et to the case where N/H is one of these nontrivial cyclic subgroups, then we have a contradiction, since the quotient N → N/H factors through the quotient N → Q; moreover, the quotient N → Q necessarily corresponds to a covering which is étale over O K because of the assumption concerning the action of Γ ⊆ Γ K on Q. Let us review what we have done so far. So far, we have: et (1) constructed quotients Π X K → Π et X K and Δ X → Δ X , as well as a section Γ K → Π et X K via various geometric considerations concerning X K ; (2) shown (when g ≥ 2) that the above quotients and section may be reconstructed entirely “group-theoretically.” Here, we pause to make the following Remark Concerning the Term “Group-Theoretic.” In [Mzk1] and [Mzk2], we imparted mathematical rigor to the term “group-theoretic” (cf. the remark on this issue in Section 14 of [Mzk2]) by specifying that it meant “preserved by isomorphism.” In the present paper, however, we would like to consider homomorphisms which are not necessarily iso- morphisms. Thus, in the present paper, when we wish to state that a certain property or object is preserved by such homomorphisms, we shall state this explicitly without using 18 the term “group-theoretic.” In the above discussion, however, we recommend the reader to simply accept this term at the level of “common sense,” since we will not use the “group- theoreticity” stated in any of the results of this Section in the proof of any of the main theorems of this paper. Finally, before continuing, we make the following important observation: Let α x : Γ K → Π X K denote the section of Π X K → Γ K determined (up to composition with an inner automorphism induced by an element of Δ X ) by a point x ∈ X K (K). Denote by et et α et x : Γ K → Π X K the composite of α x with Π X K → Π X K . Lemma 1.4. We have α et x = θ X . et Proof. It suﬃces to show that the composite of α et x with the surjection ζ X : Π X K → Δ et X is trivial. Interpreted geometrically, this simply means that “the pull-back of any étale covering Y → X (i.e., étale over O K ) to Spec(K) via x is the trivial étale covering of Spec(K).” But this assertion follows immediately from the fact that O K is strictly henselian. Section 2: Review of Galois Cohomology Let K be a p-adic ﬁeld whose residue ﬁeld is perfect. Let S → Spec(O K ) be a geometrically connected smooth morphism, and let D ⊆ S be a relative (over O K ) divisor with normal crossings. Let us write S log for the log scheme obtained by equipping S with the log structure deﬁned by D (as in [Kato]). Let us also assume that S is small (in the sense of [Falt1], [Falt2]): Recall that this simply means that S is aﬃne, say, equal to Spec(R), and, moreover, étale over some O K [X 1 , . . . , X d ] in such a way that D is schematically the inverse image of the zero locus of the function X 1 · . . . · X d . The reason we wish to deal with small (S, D) is that in [Falt1], certain Galois cohomology groups associated to such (S, D) are computed explicitly by means of the theory of almost étale extensions. The purpose of this Section is to review certain consequences of the theory of [Falt1] and [Falt2] that are of relevance to us in this paper. def Let R K = R ⊗ O K K. Let R K → R K be the maximal extension of R K which is étale (respectively, R) be the outside of D . Let R be the normalization of R in R . Let R K K def def p-adic completion of R (respectively, R); let R K = R ⊗ O K K, R K = R ⊗ O K K. Let R O K def O ) ⊗ O K. Let Γ R def be the p-adic completion of R ⊗ O O ; R = ( R = Gal(R K /R K ). K K K K K Thus, we have a natural surjection Γ R → Γ K whose kernel we denote by Δ prf R . This gives us a natural exact sequence: 1 → Δ prf R → Γ R → Γ K → 1 19 Now we would like to compute some Galois cohomology groups: Lemma 2.1. For all j ∈ Z, we have: (1) H 0 (Δ prf R , R K (j)) = R K (j); (2) H 1 (Δ prf R , R K (j)) = Ω S log /O K ⊗ R R K (j − 1). (Here the “(j)” is a Tate twist.) Proof. This follows from Theorem 4.4 of Section I of [Falt1]. Indeed, (1) follows from [Falt1], Section I, Theorem 4.4, (i) (applied to the case i = 0, and tensored with K(j)), while (2) follows from [Falt1], Section I, Theorem 4.4, (iv) (tensored with K(j)). )= R K ; (ii) H n (Γ K , R (j)) = 0 (for j = 0; Lemma 2.2. We have: (i) H 0 (Γ K , R K K )= R K . n = 0, 1); (iii) H 1 (Γ K , R K Proof. This result is due to [Tate] in the case R = O K . (Indeed, in this case, (i) and (iii) follow from [Tate], §3.3, Theorem 1; while (ii) – at least in the case j = 1 (the proof for arbitrary j = 0 is entirely similar) – follows from [Tate], §3.3, Theorem 2.) The slightly more general result stated here (i.e., for R not necessarily equal to O K ) follows by the same argument as that employed by Tate in the case R = O K (the point being that in general, R is O K -ﬂat). Alternatively, the cohomology groups in the Lemma can also be computed using almost étale extensions as in [Falt1]. Lemma 2.3. )= R (−1)) = 0. K ; (ii) H n (Γ R , R For n = 0, 1, we have: (i) H n (Γ R , R K K Proof. This follows from the Leray-Serre spectral sequence (applied to the exact sequence of groups that appears directly before Lemma 2.1), plus the preceding two Lemmas. Now let us assume that we are given an r-pointed stable log-curve f log : X log → S log of genus g, where 2g − 2 + r ≥ 1. (By this we mean that X log is obtained by pulling log log back the universal log-curve C log → M g,r via some log morphism S log → M g,r . The log structure of the universal log curve is deﬁned by the divisor with normal crossings which is the union of the marked points and the singular ﬁbers. See, e.g., [Mzk3], Section 3, for more details; cf. also the Remark preceding Deﬁnition 1.1 of this paper for a review of def the notion of a “pointed stable curve.”) Let U = S − D. Then taking the relative ﬁrst cohomology module of f| U K in the étale topology with coeﬃcients in Z p gives rise to a local system over U K , hence a Γ R -module H ∨ . As a Z p -module, H ∨ is free of rank = 2g 20 (if r = 0) and = 2g + r − 1 (if r > 0). Let H be the Γ R -module given by Hom Z p (H ∨ , Z p ). Now let def H 0 = (R 1 f ∗ O X ) ∨ (a vector bundle over S, or alternatively, a projective R-module, of rank g) and def H 1 = (f ∗ ω X log /S log ) ∨ (a vector bundle over S, or alternatively, a projective R-module, whose rank is = g if r = 0 and = g + r − 1 if r > 0). Proposition 2.4. There is a natural exact sequence (1) → H ⊗ R →H ⊗ R 0 → H 1 ⊗ R R K Z p K 0 R K → 0 which is compatible with the natural action of Γ R on all three terms. Proof. This is an immediate consequence of the “Comparison Theorem” (Theorem 6.2 of [Falt2]). The validity of the proof of this Theorem given in [Falt2] has been disputed by various mathematicians. However, since Proposition 2.4 is a relatively weak consequence of the “Comparison Theorem,” it already follows from the portion of [Falt2] that is not in dispute. (That is to say, we need only that M is “Hodge-Tate,” not that it is “crystalline.”) Alternatively, although the sort of parametrized (i.e., over a base S) Hodge-Tate decom- position that we need here is not stated in [Falt1], it follows immediately from the theory of [Falt1] by exactly the same proof as that of the main result of [Falt1]. Yet another proof of this sort of result is given in [Hyodo] (the ﬁnal Theorem – i.e., the “relative version” – in [Hyodo], §0.3), although here we need the (relatively straightforward) logarithmic generalization of [Hyodo]’s result. Section 3: The Weight Zero Quotient In this Section, we maintain the notation of the preceding Section. The purpose of this Section is to give a (rather weak) nonabelian analogue of Proposition 2.4. In fact, a much stronger “nonabelian comparison theorem” for the fundamental group of a curve can be proven, but since I do not know of any place where such a result has been written up, and, moreover, in this paper, only a relatively weak nonabelian comparison theorem is needed, I decided instead to give a rather short ad hoc treatment of this issue which will, nonetheless, be suﬃcient for the purposes of this paper. 21 log Let us denote the fundamental group of X K by Π prf X K . Thus, we have a natural prf prf surjection Π X K → Γ R . As usual, we denote the kernel of this surjection by Δ prf X ⊆ Π X K , and the maximal pro-p quotient of Δ prf X by Δ X . Moreover, by forming the quotient of prf prf Π X K by the kernel of Δ X → Δ X , we obtain Π X K . Thus, just as in Section 1, we have an exact sequence 1 → Δ X → Π X K → Γ R → 1 Moreover, any section σ : S → X of X → S whose image avoids the marked points and nodes deﬁnes a section π σ : Γ R → Π X K of the above exact sequence. Such a section π σ then deﬁnes an action of Γ R on Δ X by conjugation. Note that until one speciﬁes the section α σ : Γ R → Π X K , one only has an outer action of Γ R on Δ X (i.e., an action deﬁned only up to inner automorphisms); that is to say, in general, there is no natural action (in the usual, non-outer sense) of Γ R on Δ X until one speciﬁes a section of Π X K → Γ R . The next step is to introduce the Malčev completion of Δ X . We refer to [Del], [NT] for more details. In fact, for our purposes, it will be suﬃcient to consider a truncated form of the Malčev completion of Δ X . This truncated form may be deﬁned as follows: Let C be the category of ﬁnite dimensional Q p -vector spaces V equipped with a continuous Δ X -action that factors through Δ X /[Δ X , [Δ X , Δ X ]] and which admits a Δ X -invariant ﬁltration on whose subquotients Δ X acts trivially. (The morphisms of this category are the Q p [Δ X ]- linear morphisms V → V .) Then C is a Tannakian category over Q p , hence gives rise to an algebraic group M X over Q p . Moreover, this algebraic group M X is unipotent, hence corresponds to a nilpotent Lie algebra M X . For any Q p -algebra A, we shall write (M X ) A (respectively, (M X ) A ) for M X ⊗ Q p A (respectively, M X ⊗ Q p A). Let us write M X [1] for the commutator [M X , M X ] of this Lie algebra, and M X [0] for def the quotient M X /M X [1]. Then M X [0] may be identiﬁed with H Q p = H ⊗ Z p Q p (where H is as in the discussion following Lemma 2.3 in Section 2). Thus, we get an exact sequence 0 → M X [1] → M X → M X [0] = H Q p → 0 Moreover, the commutator [−, −] deﬁnes a surjection c X : ∧ 2 H Q p → M X [1] Now observe that although there is no natural action of Γ R on M X (unless one chooses a section of Π X K → Γ R ), there is nonetheless a natural action of Γ R on M X [0] and M X [1] with respect to which c X is equivariant. Now we have the following classical result Lemma 3.1. The kernel of c X is zero if r > 0. If r = 0, then the kernel of c X is one- ∨ dimensional, and equal to the image of the dual to the intersection form ∧ 2 H Q → Q p (−1) p (deﬁned by the cup product pairing on the cohomology of Δ X ). 22 Proof. If r > 0, then Δ X is a free pro-p group. Thus, in this case the result follows by consideration of the fact that a representation on a vector space V of the free group on n generators is the same as giving n endomorphisms of V . If r = 0, then Δ X is the quotient of a free pro-p group by a single relation; it is this relation which generates the kernel of c X . Moreover, it is easy to see from the well-known explicit form of this relation that its image in ∧ 2 H Q p is precisely as speciﬁed in the statement of the Lemma (cf. the discussion in §2 of [NT]). Next, we would like to construct a certain special quotient Z X of (M X ) . By the R K well-known categorical equivalence between unipotent algebraic groups and nilpotent Lie algebras (cf. [Del], §9), this quotient will deﬁne a quotient Z X of (M X ) . The construc- R K tion of Z X from (M X ) consists of two steps. The ﬁrst step is as follows: Consider the R K surjection → ∧ 2 H ⊗ R ∧ 2 H ⊗ Z p R K 0 R K deﬁned by projecting by means of the surjection in the short exact sequence of Proposition 2.4. It follows from Lemmas 2.3 (ii) (in the case n = 0) and 3.1 that this surjection factors through (M X [1]) . Thus, we obtain a surjection R K (M X [1]) R K → ∧ 2 H 0 ⊗ R R K By pushing forward the exact sequence 0 → M X [1] → M X → M X [0] → 0 (tensored over ) via this surjection, we thus obtain a Lie algebra U (over R ). Thus, we Q with R p K X have a surjection of Lie algebras (M X ) R K K → U X , together with an exact sequence 0 → U X [1] → U X → U X [0] → 0 def , and U [0] = (M [0]) . where U X [1] = [U X , U X ] = ∧ 2 H 0 ⊗ R R K X X R K Now we come to the second step in the construction of Z X . First let us denote by def the surjection deﬁned by the surjection in the short exact U X [0] → U X [0, 0] = H 0 ⊗ R R K sequence of Proposition 2.4. Let U X [0, 1] ⊆ U X [0] be the kernel of this surjection. Let B X ⊆ U X denote the inverse image of U X [0, 1] ⊆ U X [0] under the surjection U X → U X [0]. Then it follows from the deﬁnition of U X that B X is an abelian Lie algebra which, in fact, lies in the center of the Lie algebra of U X . (Indeed, this will follow as soon as we show the vanishing of [B X , U X ], which is equal to the image of U X [0, 1] ∧ U X [0] ⊆ ∧ 2 H Q p under the ; composite of c : ∧ 2 H → M [1] with the projection M [1] → U [1] = ∧ 2 H ⊗ R X Q p X X X 0 R K but this image is zero, by the deﬁnition of U X [0, 1].) This observation implies, in particular, that although, a priori, we have only an outer action of Γ R on U X , hence on B X , in fact, we get a natural (non-outer) action of Γ R on B X . (That is, the point is that the various 23 actions of Γ R on B X arising from diﬀerent sections of Π X K → Γ R diﬀer only by an automorphism on B X induced by conjugation by some element of Δ X ; but since B X lies in the center of U X , it follows that such an automorphism of B X is always equal to the identity.) Next, let us note that we have exact sequences 0 → B X → U X → U X [0, 0] → 0 and 0 → U X [1] → B X → U X [0, 1] → 0 where the latter exact sequence is an exact sequence of Γ R -modules (a statement which has meaning as a result of the observation at the end of the preceding paragraph). Now observe that Lemma 2.3 (ii) (for n = 0, 1 – note that here we use that U [1] = ∧ 2 H ⊗ R X 0 R K (1) is “of weight one”) implies that the is “of weight zero,” while U X [0, 1] = H 1 ⊗ R R K latter exact sequence admits a unique Γ R -equivariant splitting: B X → U X [1]. Moreover, since B X is (as observed above) contained in the center of the Lie algebra U X , it follows that the kernel of this splitting B X → U X [1] forms a Lie ideal in U X . Thus, if we then push forward the former exact sequence via this surjection B X → U X [1], we obtain a Lie algebra Z X . As usual, this Lie algebra ﬁts into an exact sequence 0 → Z X [1] → Z X → Z X [0] → 0 Moreover, one has natural identiﬁcations: Z X [1] = ∧ 2 Z X [0] (via the commutator map); (induced by the surjection of the short exact sequence of Propo- and Z [0] = H ⊗ R X 0 R K . Finally, as noted above, Z deﬁnes sition 2.4). In particular, Z X [0] is of rank g over R K X a unipotent algebraic group Z X . Deﬁnition 3.2. We shall refer to Z X as the weight zero quotient of Δ X (even though it is not literally a quotient). (Here, the “Z” of Z X stands for the “zero” of “weight zero quotient.”) Now let us ﬁx a (continuous) section α : Γ R → Π X K Then α induces an action of Γ R on Δ X . Since Z X was formed naturally – and, for that matter, group-theoretically – from Δ X , we thus obtain an action of Γ R on Z X which, in 24 general, will depend on the choice of α. The next issue we would like to consider is to what extent the sequence of Γ R -modules 0 → Z X [1] → Z X → Z X [0] → 0 admits a Γ R -equivariant section. At any rate, this exact sequence deﬁnes an extension class η α ∈ H 1 (Γ R , (Z X [0]) ∨ ⊗ Z X [1]) R K and Z [1] = By Lemma 2.3 (i) (for n = 1 – here we use that both Z X [0] = H 0 ⊗ R R K X ∧ 2 Z X [0] are “of weight zero”), it follows that this Galois cohomology group may be identiﬁed naturally with 2 (H ∨ 0 ⊗ R ∧ H 0 ) ⊗ R R K K . That is to say, one may think of η α as a section of a certain vector bundle over R Proposition 3.3. Suppose that α arises as the α σ associated to some section σ : S → X (whose image avoids the marked points and nodes – cf. the discussion at the beginning of this Section). Then η α = 0. Proof. Since everything is functorial, one reduces immediately to the universal case, as follows: In the present context, the essential data that we begin with is a r-pointed stable curve of genus g (i.e., f log : X log → S log ), plus a section (i.e., σ). The moduli stack for this data is (a certain dense open substack of) the tautological curve C → M g,r over the moduli stack M g,r of r-pointed stable curves of genus g over Z p . Thus, by “restriction,” it suﬃces to prove the Proposition in the case where S is étale over the algebraic stack C. Also, since everything involved commutes with base-extension, it is easy to see that we may assume that k is algebraically closed. Then over some dense open T ⊆ (S − D) ⊗ O K k, the Jacobian of X ⊗ O K k will be ordinary. Thus, every point of β ∈ S(O K ) that maps Spec(k) ⊆ Spec(O K ) into T deﬁnes an ordinary (in the sense of Deﬁnition 1.1) hyperbolic curve Y β → Spec(K) over K (by restricting (the complement of the marking sections in) X → S to β). Next, observe that that if we restrict the section σ to the point β, we get a section et σ β : Spec(K) → Y β which induces an action of Γ K on Δ et Y β (i.e., the quotient Δ Y β → Δ Y β considered in Section 1). By Lemmas 1.2 and 1.4, this action is, in fact, the trivial action. Now I claim that in the case of such an ordinary curve: The quotient Δ Y β → Δ et Y β (considered in Section 1) induces a natural (in particular, Γ K -equivariant) isomorphism of the weight zero quotient 25 Z Y β of Δ Y β (= Δ X ) with the Malčev completion of Δ et Y β (truncated at the second step and tensored over Z with K). p Indeed, it follows immediately from the construction of Z Y β that the surjection Δ Y β → Δ et Y β induces a morphism from Z Y β to the “weight zero quotient” (i.e., constructed in a fashion analogous to the construction of Z Y β ) of Δ et Y β . On the other hand, since the action of et Γ K on Δ Y β is trivial, it follows that the “weight zero quotient” of Δ et Y β is just the Malčev Moreover, completion of Δ et (truncated at the second step and tensored over Z with K). p Y β it is clear that the map induced on abelianizations by this morphism from Z Y β to this weight zero quotient of Δ et Y β is an isomorphism. (Indeed, surjectivity follows from the fact that we are dealing (by deﬁnition) with various quotients of a single object; injectivity then follows from Proposition 2.4 and Lemma 2.3 (ii) (for n = 0), and the fact that both Thus, (since Δ et is a free pro-p group of rank g abelianizations are of rank g over K.) Y β – cf. the discussion following Deﬁnition 1.1) we conclude that we get an isomorphism as stated in the “claim.” This completes the veriﬁcation of the claim. Thus, any minimal choice of generators of Δ et Y β (which will necessarily be ﬁxed by Γ K ) deﬁnes a splitting of the sequence of Γ K -modules 0 → Z X [1] → Z X → Z X [0] → 0 Here the Γ K -action is given by composing the Γ R -action considered above with the mor- phism – well-deﬁned up to composition with an inner automorphism (which does not bother us since this inner automorphism corresponds to a coboundary in the computation of the cohomology class η α that we are interested in) – Γ K → Γ R deﬁned by β. But from the deﬁnition of the extension class η α , this means that the restriction 2 η α | β ∈ (H ∨ 0 ⊗ R ∧ H 0 ) ⊗ R,β K of η α to the point β is zero. On the other hand, if η α is zero when restricted to any such β (i.e., any β ∈ S(O K ) that maps Spec(k) ⊆ Spec(O K ) into T ), it is clear that η α itself must be zero. This completes the proof of the Proposition. Note that so far, in this Section and the last, we have been dealing with families of curves, parametrized over a base S. Before continuing on to the next Section, it is worthwhile to go back to the case of “a single curve” over K in order to make explicit the consequences for such single curves of the theory developed thus far. Thus, let X K → Spec(K) be a hyperbolic curve. As in Section 1, we have an exact sequence 1 → Δ X → Π X K → Γ K → 1 26 Then let us ﬁrst note that the theory reviewed in Section 2 can be applied in the present nonparametrized situation as well to produce (following precisely the same recipe as in the parametrized situation considered as above) a Lie algebra Z X , together with a group Z X (both over K). Deﬁnition 3.4. We shall refer to Z X as the weight zero quotient of Δ X . Let x ∈ X K (K). Then x deﬁnes a section α x : Γ K → Π X K (well-deﬁned up to composition with an inner automorphism arising from Δ X ). Moreover, α x deﬁnes an action of Γ K on Z X . Proposition 3.5. of Γ K -modules Relative to the action of Γ K on Z X deﬁned by α x , the exact sequence 0 → Z X [1] → Z X → Z X [0] → 0 splits. Proof. Note that this Proposition is not, strictly speaking, a special case of Proposition 3.3 (i.e., where we take S = O K ), since Proposition 3.3 only addresses the case where the divisor of bad reduction is ﬂat over O K . However, by means of specialization, one can reduce the present Proposition to the “universal case” considered during the proof of Proposition 3.3. Moreover, in this case, Proposition 3.3 already tells us that the relevant exact sequence is split. This completes the proof. Remark. Let us write (Δ X ) Q p for the full (i.e., not truncated as above) Malčev completion (cf. [Del], [NT]) of Δ X . Then in some sense, Proposition 3.5 above is a truncated version of a theorem that states “relative to the action of Γ K on (Δ X ) Q p deﬁned by α x , (Δ X ) Q p is Hodge-Tate.” Here, since (Δ X ) Q p is an inverse limit of unipotent algebraic groups, one can interpret “Hodge-Tate” to mean that the Lie algebras of each of these unipotent algebraic groups are Hodge-Tate representations. In fact, it is not diﬃcult to prove that (Δ X ) Q p is Hodge-Tate (even for arbitrary higher-dimensional smooth X K → Spec(K)) as follows: One reduces the higher-dimensional case to the curve case by cutting with hyperplane sections. Then for curves, by considering the universal case, one can reduce to the case of curves smooth over O K . Finally, for curves smooth over O K , one can apply the techniques of [Falt1] (by considering cohomology spaces with coeﬃcients valued in unipotent algebraic groups over Q p ). This shows that (Δ X ) Q p is Hodge-Tate. Moreover, one can also construct ∞ a nontruncated “weight zero quotient (Δ X ) Q p → Z X ” (that is to say, if X K is a curve of ∞ genus g, then Z will be (Γ -equivariantly) isomorphic to the Malčev completion over K X K of the free group on g generators). In fact, it is even possible to show that (Δ X ) Q p is de Rham, but we shall not pursue such issues here since they are not relevant to the proof of 27 the main results of this paper. Nevertheless, we remark that this sort of theorem has been veriﬁed by A. Shiho ([Shiho]), in a manuscript in preparation. Section 4: J-Geometric Sections We maintain the notations of the latter portion of Section 3. Moreover, we assume that our curve X K → Spec(K) is proper (hence of genus ≥ 2) and that X K (K) = ∅. In this Section, we would like to consider a continuous homomorphism α : Γ K → Π X K which deﬁnes a section of the surjection Π X K → Γ K . The section α deﬁnes an action of Γ K on Δ X by conjugation, and hence also an action of Γ K on Z X . In particular, we would like to consider the signiﬁcance of the following condition on α: (∗) spl The exact sequence of Γ K -modules (relative to the action deﬁned by α) 0 → Z X [1] → Z X → Z X [0] → 0 splits. The main result of this Section is to show that the group-theoretic condition (∗) spl is equivalent to the statement that α is “J -geometric” (a term which means that α acts in some respects as if it came from a geometric point x ∈ X K (K) – see Deﬁnition 4.3 for a precise deﬁnition). To do this, ﬁrst we need to recall certain facts concerning Jacobians and their funda- (d) mental groups. For d ∈ Z, let J X → Spec(K) be the Picard scheme of line bundles on X K (d) (0) (d) of degree d. Thus, J X is a torsor over J X = J X , the Jacobian of X K . Note that J X is deﬁned even if X K does not admit any K-rational points, as the scheme representing the étale sheaﬁﬁcation of the usual Picard functor of degree d line bundles. Note that a base- point x ∈ X K (Ω) (where Ω is the algebraically closed ﬁeld of Section 0) may be regarded as a degree one divisor on X Ω = X K ⊗ K Ω, hence (by multiplying this divisor by d) we (d) obtain a point x d ∈ J X (Ω). This allows us to deﬁne the arithmetic fundamental group (d) (d) π 1 (J X , x d ), as well as its geometric counterpart π 1 (J X , x d ). If we form the quotient (d) K (d) of π 1 (J X , x d ) by the kernel of π 1 (J X , x d ) → Δ (d) (i.e., the projection to the maximal J X K pro-p quotient of the geometric fundamental group), then we obtain a topological group Π (d) . Moreover, we have a natural exact sequence J X 28 1 → Δ J (d) → Π J (d) → Γ K → 1 X X (1) Finally, the natural embedding X K → J X induces a natural surjection Π X K → Π (1) , J X whose kernel is the commutator subgroup of Δ X . Now let us observe that we can reconstruct Π (d) from Π X K → Γ K , as follows. First of J X all, we can reconstruct Π (1) as the quotient of Π X K by the commutator subgroup of Δ X . J X (d) Next, let us observe that, since (for all d ∈ Z) J X is a J X -torsor, considering the action (d) of J X on J X allows one to identify Δ (d) with Δ J X . Thus, we have an exact sequence J X 1 → Δ J X → Π J (1) → Γ K → 1 X (1) (d) If we consider (for nonzero d), the morphism J X → J X given by multiplication by d, we d see that the result of pushing forward this sequence by means of the morphism Δ J X → Δ J X (i.e., multiplication by d) gives rise to an exact sequence which can be naturally identiﬁed with the exact sequence 1 → Δ J (d) = Δ J X → Π J (d) → Γ K → 1 X X This completes our review of Jacobians and their fundamental groups. Now recall from the deﬁnition of Z X [0] in Section 3 that Z X [0] may be identiﬁed with In particular, one has a natural Γ -equivariant the weight zero portion of Δ ⊗ K. J X Z p K morphism Δ J X → Z X [0]. Thus, by pushing forward the exact sequence of the preceding paragraph (for d = 1) by means of this morphism, we obtain a morphism of exact sequences of topological groups as follows: Δ J X ⏐ ⏐ −→ 1 −→ Z X [0] −→ 1 −→ Π (1) J X ⏐ ⏐ −→ Γ K ⏐ ⏐ −→ Π Z (1) −→ Γ K −→ J X 1 1 1 Next, we would like to consider sections of Π (1) → Γ K and Π Z (1) → Γ K . Recall that the J X J X diﬀerence between any two sections of Π (1) → Γ K (respectively, Π Z (1) → Γ K ) is given by J X J X an element of H 1 (K, Δ J X ) (respectively, H 1 (K, Z X [0])). Moreover, if θ L , θ M : Γ → Π (1) (1) are sections that arise from geometric points L, M ∈ J X (K), then the diﬀerence def δ = θ L − θ M ∈ H 1 (K, Δ J X ) 29 J X maps to zero in H 1 (K, Z X [0]) ([BK], Example 3.11 – cf. the proof of Lemma 4.1 for some more details). Thus, it follows that the section θ L [0] : Γ K → Π Z (1) obtained from θ L J X by composing θ L with Π (1) → Π Z (1) is independent (up to composition with an inner J X J X automorphism of Π Z (1) arising from Z X [0] ⊆ Π Z (1) ) of the choice of L. In particular, as J X J X (1) long as J X (K) is nonempty (which is the case here, since X K (K) has been assumed to be nonempty), we thus obtain a canonical section (well-deﬁned up to composition with an inner automorphism of Π Z (1) arising from Z X [0] ⊆ Π Z (1) ) J X J X θ geom : Γ K → Π Z (1) J X of Π Z (1) → Γ K . J X Now let θ : Γ K → Π (1) be an arbitrary section, and let θ[0] : Γ K → Π Z (1) denote J X J X the section induced by θ. Then taking the diﬀerence between θ[0] and θ geom deﬁnes a cohomology class δ θ ∈ H 1 (K, Z X [0]) = H 0 (where the last equality follows from Lemma 2.3, applied in the case R K = K). Now we have the following result: Lemma 4.1. Suppose that the residue ﬁeld k is ﬁnite. Then there exists a geometric (1) point L ∈ J X (K) such that θ = θ L if and only if δ θ = 0. Proof. This follows immediately from the theory of [BK], §3, especially Example 3.11, plus the following observation: If H is (as in Section 3) the Γ K -module that arises as the abelianization of Δ X , then + Ker{H 1 (K, H) → H 1 (K, H ⊗ B DR )} (i.e., by Lemma 3.8.1 of [BK], “H g 1 ” in the notation of [BK]) is equal to = H 1 (K, Z [0])} Ker{H 1 (K, H) → H 1 (K, H ⊗ K) X (where the last equality follows from Proposition 2.4 and Lemma 2.2 (ii) (for n = 1, j = 1)). + Indeed, this equality of kernels follows by using the natural ﬁltration on B DR (whose i ≥ 0), together with the fact that H 1 (K, H ⊗ K(i)) =0 subquotients are equal to K(i), for i > 0 (by Proposition 2.4, Lemma 2.2 (ii) (for n = 1, j > 0)). 30 Now, we return to considering the section α : Γ K → Π X K . Let β : Γ K → Π X K be any section that arises from some geometric point x ∈ X K (K). We would like to compare α and β, and, in particular, show that, under the assumption (∗) spl , α and β are, in some sense (to be speciﬁed precisely below), relatively close. Let ζ : Γ K → Δ X be the continuous function (not necessarily a group homomorphism) such that α(γ) = ζ(γ) · β(γ) ∈ Π X K for all γ ∈ Γ K . (respectively, If γ ∈ Γ K , then let us write γ α (φ) ∈ Z X ( K) Now let φ ∈ Z X ( K). for the result of letting γ act on φ by means of the action deﬁned by α γ (φ) ∈ Z ( K)) β X (respectively, β). Note that it follows from the construction of Section 3 that we have a For ∈ Δ , let us denote by the image of natural morphism Δ → Z ( K). ∈ Z ( K) X X Then we have the following: in Z X ( K). Lemma 4.2. X Z X We have: γ α (φ) = ζ(γ) Z · γ β (φ) · ζ(γ) −1 Z . Proof. Indeed, this follows immediately from the fact that the respective actions of Γ K are induced by conjugation by α(γ) and β(γ) inside Π . on Z X ( K) X K Let α : Γ K → Π X K be any section of Π X K → Γ K , and let α Z : Γ K → Π Z (1) be the section obtained by composing α J X Z with Π X K → Π (1) . Then we make the following J X Deﬁnition 4.3. We shall call α J -geometric if α Z coincides with θ geom (up to compo- sition with an inner automorphism of Π Z (1) arising from Z X [0] ⊆ Π Z (1) ). J X J X The following result is the main technical observation that made it possible to substantially strengthen the result of [Mzk2]. Proposition 4.4. Let α : Γ K → Π X K be a continuous group homomorphism that deﬁnes a section of Π X K → Γ K . Then α satisﬁes (∗) spl if and only if α is J -geometric. be β-invariant, i.e., invariant under the action of Γ on Z ( K) Proof. Let φ ∈ Z X ( K) K X deﬁned by β. We would like to calculate the action of Γ K on φ that is induced by α. Thus, for γ ∈ Γ K , we have, by Lemma 4.2: γ α (φ) = ζ(γ) Z · γ β (φ) · ζ(γ) −1 Z = ζ(γ) Z · φ · ζ(γ) −1 Z 31 Thus, γ α (φ) · φ −1 = [ζ(γ) Z , φ] (where the brackets denote the “commutator”). Note that = Z [1] = ∧ 2 Z [0] = (∧ 2 H ) ⊗ K [ζ(γ) Z , φ] ∈ Z X [1]( K) 0 K X X Now it remains to reinterpret the calculation just performed in terms of cohomology classes. Recall the class 2 2 η α ∈ H ∨ 0 ⊗ K ∧ H 0 = Hom K (H 0 , ∧ H 0 ) in Z [0]( K) = Z [0]. Since discussed in Section 3. Let φ[0] be the image of φ ∈ Z X ( K) X X φ is β-invariant, it follows that φ[0] is a Γ K -invariant element of Z X [0], hence belongs to = Z [0]. Let δ def = α −θ ∈ H 1 (K, Z [0]) = H . Then it is immediate H ⊆H ⊗ K 0 0 K X α Z geom 0 X from the deﬁnitions that the calculation of the preceding paragraph, interpreted in terms of cohomology classes, becomes: η α (φ[0]) = δ α ∧ (φ[0]) (Note that since θ geom may be computed – cf. the discussion preceding Lemma 4.1 – using the geometric section β, it follows that δ α is precisely the cohomology class deﬁned by the in Z [0]( K) = Z [0].) Next observe that this image of the cocycle ζ(−) Z : Γ K → Z X ( K) X X = Z [0] = equation holds for all β-invariant φ, and that every element of H ⊆ H ⊗ K 0 0 K X lifts to a β-invariant φ ∈ Z ( K). Z X [0]( K) (Indeed, this follows from Proposition 3.5 X since β arises from a point of X K (K).) Thus, it follows that η α : H 0 → H 0 ∧ H 0 is simply the map “δ α ∧.” In particular, η α = 0 if and only if δ α = 0. This completes the proof of the Proposition. Remark. So far here we have been dealing with the truncated weight zero quotient Δ X → Z X , but it would be interesting also to see what happens in the case of the full nontruncated ∞ weight zero quotient Δ X → Z X (as in the Remark at the end of Section 3). For instance, ∞ ∞ if the action of α on Z X is such that Z X has “enough invariants” (i.e., there exists a pro- is Γ -equivariantly isomorphic to Z ∞ ), does algebraic group G over K such that G ⊗ K K K X it follow that α itself automatically comes from a geometric point x ∈ X K (K)? Although such questions are beyond the scope of this paper, it is the opinion of the author that such questions deserve further study. Note that this sort of issue is closely related to the so-called Section Conjecture – cf. the Remark following Theorem 19.1 for more details on this conjecture. 32 Section 5: The J-Geometricity of K-Valued Points Let K be a p-adic ﬁeld with perfect residue ﬁeld. Let X K and Y K be proper hyperbolic curves over K. Let U K be the K-scheme obtained by localizing X K at its generic point. Thus, the underlying topological space of U K consists of one point, and the ring of functions on U K is the function ﬁeld K X of X K . Note in particular that we can consider Π U K , Π Y K . In particular, Π U K is a certain quotient of the absolute Galois group of K X . Let us assume that we are given a continuous open homomorphism θ : Π U K → Π Y K over Γ K . In this Section, we would like to begin the proof of the main theorem of this paper by showing that any such θ necessarily “maps geometric sections to J -geometric sections.” (Naturally, we will explain below precisely what is meant by the expression in quotes.) First observe that if x ∈ X K (K) is any K-valued point, then we can form the com- pletion (K X ) x of the ﬁeld K X with respect to the valuation deﬁned by x. Moreover, we have a natural morphism Π (K X ) x → Π U K (well-deﬁned up to composition with conjugation by an element of Δ U ) whose image is “the” (more rigorously: any of the various conju- gate) decomposition group associated to x. Moreover, as is well-known (see, e.g., [Ser2]), (K X ) x is (noncanonically) isomorphic to K((t)) (where t is an indeterminate), so Δ (K X ) x may be identiﬁed with Z p (1). In particular, by forming, relative to some isomorphism (K X ) x ∼ = K((t)), the ﬁeld extension of (K X ) x corresponding to adjoining a compatible system of p-power roots of t to K((t)), one sees immediately that Π (K X ) x → Γ K admits many sections. Deﬁnition 5.1. We shall refer to as geometric any section Γ K → Π U K of Π U K → Γ K obtained by composing a section of Π (K X ) x → Γ K with (any one of the conjugate natural homomorphisms) Π (K X ) x → Π U K . Note, in particular, that Π (K X ) x → Π X K factors through Γ K , so in particular, the com- posite α X : Γ K → Π X K with Π U K → Π X K of any geometric section α U : Γ K → Π U K is induced by some point x : Spec(K) → X K . Let α U : Γ K → Π U K be a geometric section. By composing α U with θ, we obtain a section α Y : Γ K → Π Y K . In this Section, we would like to prove that α Y is necessarily J -geometric in the sense of Deﬁnition 4.3. To do this, let us ﬁrst observe that we have a diagram of continuous morphisms Δ U −→ ⏐ ⏐ θ| Δ U Z X ( K) −→ Z Y ( K) Δ Y 33 which are compatible with the Γ K -actions deﬁned by α U and α Y on the all the groups involved. (Here, the upper horizontal morphism is obtained by composing Δ U → Δ X with Δ → Z ( K).) Then we have the following X X → Z ( K) Lemma 5.2. There exists a (natural) surjective Γ K -homomorphism Z X ( K) Y that makes the above diagram commute. Proof. Observe that the kernel of Δ U → Δ X is generated by inertia groups (i.e., images of the various Δ (K X ) x → Δ U ), and the action of Γ K on an inertia group is via the cyclotomic character, i.e., (in the language of Hodge-Tate Galois representations) of weight one. It thus follows (from Lemma 2.2 (ii), for n = 0, j = −1) that the morphism Δ → Z ( K) U Y (obtained from the diagram above) factors through Δ X . Hence, we obtain a morphism But now, it follows immediately (by the universal property of the truncated Δ → Z ( K). X Y from Δ Malčev completion, plus “weight arguments”) from the construction of Z X ( K) X factors naturally through Z ( K). that this morphism Δ → Z ( K) This shows the X Y X existence of a morphism as claimed in the statement of the Lemma. The fact that this morphism is surjective follows from the fact that θ is open (which implies that the induced ab morphism Δ ab U → Δ Y on abelianizations is open, hence surjective after tensoring with Q p ). deﬁned by α satisﬁes the condition (∗) spl Lemma 5.3. The action of Γ K on Z Y ( K) Y discussed in Section 4. deﬁned by α satisﬁes (∗) spl follows Proof. Indeed, that the action of Γ K on Z X ( K) X from Proposition 3.5. Thus, Lemma 5.3 follows from the surjectivity of the morphism of Lemma 5.2. Proposition 5.4. Let θ : Π U K → Π Y K be a continuous open homomorphism over Γ K , α U : Γ K → Π U K a geometric (Deﬁnition 5.1) section, and α Y : Γ K → Π Y K the composite of α U with θ. Then α Y is J -geometric (Deﬁnition 4.3). Proof. This follows by combining Lemma 5.3 with Proposition 4.4. Section 6: F-Geometricity and FI-Geometricity def Let K be a ﬁnite extension of Q p . Let S = Spec(R), where R is an O K -algebra noncanonically isomorphic to O K [[t]] (and t is an indeterminate). Let η S be the generic 34 def def point of S (regarded as a scheme). Let Γ S = π 1 (S K ) (where S K = S ⊗ O K K, and the “π 1 ” is with respect to some base-point which we omit to simplify notation); let Γ η S be the absolute Galois group of K(η S ) (the function ﬁeld of S). Thus, Γ S can be naturally regarded as a quotient of Γ η S . If H ⊆ Γ η S is an open subgroup corresponding to some ﬁnite étale covering η S → η S , then we shall write Γ η S for H; S = Spec(R ) for the normalization of S in η S ; and Γ S for the quotient of Γ η S corresponding to étale coverings def of S K = S ⊗ O K K. Thus, S is ﬁnite and ﬂat (since S is regular of dimension 2, and R is normal, hence an R-module of depth 2) over S. Remark. For the reader familiar with [Mzk2], we remark that this Section and the next are, in some sense, a generalization of Section 5 of [Mzk2], and play a comparable role in the present paper to that of Section 5 in [Mzk2]. For the reader not familiar with [Mzk2], we remark that, nevertheless, we do not assume any knowledge of [Mzk2] in the following discussion. We begin by considering a continuous Z p [Γ η S ]-module V , where V , as a Z p -module, is a ﬁnite and free. Deﬁnition 6.1. We shall refer to V as potentially geometric if there exists some open subgroup Γ η S ⊆ Γ η S such that the Γ η S -module obtained by restricting the Γ η S -action on V to Γ η S arises as the Tate module of some p-divisible group G → Spec(O K ), where K ⊆ R K is a ﬁnite extension of K. Note, in particular, that the Γ η S -action on V then factors through the quotient Γ η S → Γ S . Moreover, if V is potentially geometric, then we may make the following construction: Let us write G for the (not necessarily connected) formal group (over O K ) associated to the p-divisible group G → Spec(O K ) of Deﬁnition 6.1. Thus, we have a natural isomorphism (of ﬁnite ﬂat group schemes over O K ) between the kernels G[p n ] ∼ = G[p n ] of multiplication by p n (for all n ≥ 0) on G and G. In particular, if we consider the exact sequence (generalizing the Kummer sequence, which corresponds to the case where G is the formal group associated to the multiplicative group G m ) 0 −→ G[p n ] −→ G p n −→ G −→ 0 as an exact sequence of sheaves on the ﬁnite ﬂat site of S , then we get a natural map 1 (S , G[p n ]) G(S ) → H ﬂat 35 (where the cohomology group is relative to the ﬁnite ﬂat topology on S ) for n ≥ 0. Since étale morphisms are quasi-ﬁnite and ﬂat, we also have a natural morphism 1 1 H ﬂat (S , G[p n ]) → H et (S , G[p n ] K ) Thus, if we compose the above two morphisms, take the inverse limit with respect to n, 1 and use the fact that H et (S , −) ∼ = H 1 (Γ S , −), we get a natural morphism κ G : G(S ) → H 1 (Γ S , V ) which one may regard as a generalization of the Kummer map (from units of a ﬁeld to a certain Galois cohomology group of the ﬁeld). Remark. Note that when the formal group G arises from an abelian variety over O K , then the cohomology class that one obtains by applying κ G to a point of G coincides with the cohomology class that one obtains (cf. the discussion of Section 4) by looking at the morphism induced on arithmetic fundamental groups by the corresponding point of the abelian variety. Indeed, this is a matter of general nonsense – cf., e.g., [Naka2], Claim (2.2); [NTs], Lemma (4.14). Deﬁnition 6.2. Suppose that V is potentially geometric, and that Γ η S ⊆ Γ η S is as in Deﬁnition 6.1. Then we deﬁne def H f 1 (S K , V ) ⊆ H 1 (S K , V ) = H 1 (Γ S , V ) to be the subset of elements ζ ∈ H 1 (S K , V ) such that some nonzero multiple of ζ lies in the image of G(S ) under the morphism κ G . Now let X η S → η S be a proper hyperbolic curve over η S such that for some open Γ η S ⊆ Γ η S , and some proper, hyperbolic curve Z → Spec(K ) (where K ⊆ R K is a ﬁnite extension of K, and we assume that Z(K ) = ∅), we have an isomorphism of η S -curves X η S × η S η S ∼ = Z × K η S Then we make the following technical Deﬁnition 6.3. We shall call X η S irreducibly splittable if for some open Γ η S ⊆ Γ η S , and some proper, hyperbolic curve Z → Spec(K ) (where K ⊆ R K is a ﬁnite extension of K, and we assume that Z(K ) = ∅), we have an isomorphism of η S -curves X η S × η S η S ∼ = Z × K η S 36 and, moreover, the (induced) map S → Spec(O K ) satisﬁes: (i) S → Spec(O K ) admits a section; (ii) the morphism S → Spec(O K ) has geometrically irreducible ﬁbers. As usual, we have an exact sequence 1 → Δ X η = H X → Π X η → Γ η S → 1 S S where we write H X for the abelianization of Δ X η . Thus, H X has a natural structure of S continuous Γ η S -module. Moreover, as a Z p -module, it is free of rank 2g X (where g X is the genus of X η S ). Let us write I H X → H X for the quotient of H X by all elements of H X on which some open subgroup of Γ η S acts via I the cyclotomic character. (The “I” comes from the fact that H X is obtained by forming the quotient of H X by all of its “inertia-like” subgroups.) Next, let F ⊆ H X H X F be the largest Γ η S -submodule of H X which has no nonzero torsion-free quotients H X → Q such that some open subgroup of Γ η S acts trivially on Q. (Here, the “F ” stands for F “ﬁnite.” This is because H X corresponds to the portion of H X that (potentially) extends to a p-divisible group – i.e., a direct limit of ﬁnite ﬂat group schemes – over O K .) Then F I it is well-known (see, e.g., [FC], Chapter III) that H X and H X are Cartier-dual to one F another, and, moreover, that H X is potentially geometric (Deﬁnition 6.1). (Indeed, in [FC], Chapter III, one ﬁnds a discussion of how one may obtain (in a natural fashion) abelian varieties over K as “quotients” (by some group of periods) of semi-abelian varieties that (potentially) extend over O K in such a way that the dimensions of their toral parts are the same over the generic and special points of O K . If we apply this theory to the Jacobian J Z F of the curve Z, then H X is the Tate module of this semi-abelian variety (that potentially I F I is the Cartier dual of H X , it thus follows that H X is also extends over O K ).) Since H X potentially geometric. Let FI I H X ⊆ H X F I F FI FI be the image of H X in H X . Thus, we have a surjection H X → H X , and H X is also potentially geometric. Finally, let def M I F H X = Ker(H X → H X ) ⊆ H X 37 M (Here, the “M” stands for “multiplicative.” This is because H X arises from the portion F that corresponds to twisted copies of (the Tate module of) the multiplicative group of H X M G m .) H X is also potentially geometric. Next, we would like to consider a section α : Γ η S → Π X η of Π X η → Γ η S . Now recall S S the isomorphism X η S × η S η S ∼ = Z × K η S By composing α| Γ η : Γ η S → Π X η with the projection Π X η → Π Z induced by this S S S isomorphism, we thus obtain a morphism Γ η S → Π Z , whose composite with Π Z → Γ K is the natural morphism Γ η S → Γ K . Now let us make the following assumption on α: (∗) S This morphism Γ η S → Π Z factors through Γ S . Let us denote the resulting morphism by β : Γ S → Π Z . Now let γ : Γ S → Π Z be any morphism obtained by composing the natural morphism Γ S → Γ K with some section Γ K → Π Z arising from a geometric point ∈ Z(K ). Let β J , γ J : Γ S → Π (1) be the morphisms obtained by composing β and γ, respectively, with J Z Π Z → Π (1) . Then the diﬀerence β J − γ J deﬁnes an element J Z , H Z ) δ Z ∈ H 1 (S K hence an element , H X ) δ X ∈ H 1 (S K I I whose image in H 1 (S K , H X ) we denote by δ X . Deﬁnition 6.4. Suppose that α : Γ η S → Π X η is a section that satisﬁes the assump- S tion (∗) S above. Then we shall call α F -geometric (respectively, F I-geometric) if some I F nonzero multiple of δ X (respectively, δ X ) lies in the image of H f 1 (S K , H X ) (respectively, 1 FI 1 1 I H f (S K , H X )) in H (S K , H X ) (respectively, H (S K , H X )). Note that if α is F -geometric, it is also automatically F I-geometric. Also, let us observe that the deﬁnition of “F -geometric” or “F I-geometric” is independent of the choice of γ (cf. Lemma 4.1 and the discussion preceding it). Indeed, to see this, it suﬃces to verify that some nonzero multiple of the class ∈ H 1 (S K , H X ) arising from the diﬀerence of two 1 F γ’s lies in H f (S K , H X ). But the diﬀerence of two γ’s (both of which arise from a geometric point ∈ Z(K )) deﬁnes a point ∈ J Z (K ) (where J Z is the Jacobian of Z). Moreover, since the residue ﬁeld of K is ﬁnite, it follows that some nonzero multiple of this point in J Z (K ) 38 extends to an O K -valued point of the Néron model J Z of J Z over O K which maps the special point of Spec(O K ) to the identity of J Z . On the other hand, since G F – i.e., the F “G” (cf. the discussion following Deﬁnition 6.1) for the potentially geometric module H X – is simply the formal group deﬁned by J Z , it thus follows that such a point ∈ J Z (O K ) deﬁnes a point in G F (O K ) ⊆ G F (S ). Thus, by Deﬁnition 6.2, we see that the diﬀerence F between the two γ’s lies in H f 1 (S K , H X ), as desired. Proposition 6.5. Suppose that X S is irreducibly splittable (cf. Deﬁnition 6.3). Then any F I-geometric α is also F -geometric. Proof. First, let us state that throughout the proof, “S ” will be a ﬁxed S satisfying the conditions of Deﬁnition 6.3. Next, let us observe that the cokernel of the natural morphism F FI H f 1 (S K , H X ) → H f 1 (S K , H X ) is torsion. Indeed, this follows from Deﬁnition 6.2 and the fact that if G F and G F I are the respective formal groups as in Deﬁnition 6.2, then the natural morphism G F → G F I is formally smooth (hence surjective on S -valued points). I , H X ) whose image ζ I ∈ H 1 (S K , H X ) lies Thus, if we start with a class ζ ∈ H 1 (S K 1 FI in the image of H f (S K , H X ), then (after replacing ζ by a nonzero multiple of ζ), we , H X ) such that: (a.) ζ lies in the image may assume that there exists a ζ ∈ H 1 (S K 1 F 1 I of H f (S K , H X ); (b.) ζ − ζ maps to 0 in H (S K , H X ). Moreover, by the deﬁnition of def M I H X = Ker(H X → H X ), it follows that (b.) may be rewritten in the form: “ζ − ζ lies in M the image of H 1 (S K , H X ) in H 1 (S K , H X ).” With these observations in hand, it follows that it suﬃces to prove that M (∗) im The image of H 1 (S K , H X ) in H 1 (S K , H X ) is contained up to torsion (i.e., up to multiplication by a nonzero integer) in the image of F , H X ) in H 1 (S K , H X ). H f 1 (S K Note ﬁrst that by replacing K by a ﬁnite extension of K (and thus also enlarging S – note that this does not aﬀect the validity of the conditions (i) and (ii) of Deﬁnition 6.3), M we may assume that the action of Γ S on H X (−1) (where the “(−1)” is a Tate twist) is M trivial. (Thus, in particular, as a Γ S -module, H X is isomorphic to a direct sum of a ﬁnite number of copies of Z p (1).) On the other hand, by Lemma 6.6 below (and the Kummer , Z p (1)) = {(R K ) × } ∧ (where the “∧” denotes p-adic exact sequence), it follows that H 1 (S K , Z p (1)) = {(R ) × } ∧ and (the image of) completion) is generated up to torsion by H f 1 (S K M , H X ) is generated up to torsion H 1 (K , Z p (1)) = {(K ) × } ∧ . Thus, it follows that H 1 (S K 1 M 1 M by H f (S K , H X ) and (the image of) H (K , H Z ). Next, observe that the image of H 1 (K , H Z M ) in H 1 (K , H Z ) is contained up to torsion in the image of H f 1 (K , H Z F ) in H 1 (K , H Z ). Indeed, this follows from the theory of [BK], 39 §3 (cf. the proof of Lemma 4.1). Namely, since H Z M is “of weight one,” the image of is zero, so any cohomology classes H M ⊆ H in the weight zero component of H ⊗ K Z Z 1 Z But M 1 (K , H Z ) go to zero in H (K , −) of the weight zero component of H Z ⊗ Z p K. in H this means (as we saw in Lemma 4.1), that such cohomology classes arise geometrically, as desired. (Note that, by the above argument involving Néron models (in the discussion immediately preceding Proposition 6.5), it follows that (relative to pulling back cohomology classes over K to cohomology classes over S K ), the notation “H f 1 (S K , −)” of Deﬁnition 1 6.2 is consistent with the notation “H f (K , −)” of [BK], §3.) Putting everything together, we thus see that (∗) im has been veriﬁed. This completes the proof of the Proposition. Lemma 6.6. For S satisfying condition (ii) of Deﬁnition 6.3, we have that (R K ) × is generated up to torsion by (R ) × and (K ) × . Proof. Indeed, condition (ii) of Deﬁnition 6.3 implies that if π is a uniformizing element of O K ⊆ R , then the ideal π · R is contained in a unique prime ideal of height one of R . Let us denote this prime ideal by ℘. Let x ∈ R be an element which becomes a unit in R K . Then x is invertible at every height one prime of R except (possibly) ℘. Moreover, by replacing x by some x n (where n is independent of x), we may assume that x has the same valuation as π m (for some nonnegative integer m) in the discrete valuation ring R ℘ . It thus follows that π −m · x is a unit at every height one prime of R . Since R is normal, this implies that π −m · x ∈ (R ) × . This completes the proof of the Lemma. Section 7: From F-Geometricity to Line Bundles In this Section, we use some elementary algebraic geometry (Lemmas 7.1 and 7.2) to translate the rather abstract and technical condition of “F -geometricity” into a more tractable existence criterion (Proposition 7.4) for line bundles. Let S be the spectrum of a ﬁeld of characteristic zero. Let X → S be a proper hyperbolic curve of genus g over S. Let (N ) N be a ﬁxed positive integer. We would like to consider the natural morphism X → J X (given by mapping a point x of X to the line bundle O X (N · x)). Taking the product of (N ) this morphism with X (on the right), we obtain a morphism ξ : X × S X → J X × S X. Lemma 7.1. (N ) There exists a line bundle on J X × S X whose pull-back via ξ is a nonzero def tensor power of the line bundle D = O X× S X (Δ) (where Δ ⊆ X × S X) on X × S X. Proof. First, let us take the product of ξ with one more copy of X on the right, to (N ) obtain a morphism ξ : X × S X × S X → J X × S X × S X. For i, j = 1, 2, 3 such that 40 i = j, let Δ ij ⊆ X × S X × S X denote the diagonal given by setting equal the i th and j th components of the triple product. Let D ij denote the line bundle O X× S X× S X (Δ ij ). In the triple product of X, we consider the second X to be the “true curve X”; the ﬁrst X to be a “parameter space for the family of line bundles D 12 ” (regarded as line bundles on the “true curve” given by the second factor); and the third X to be a “base extension X → S.” Over this extended base, the “true curve” acquires a “section” Δ 23 . Thus, it follows from the general theory of the Picard functor of a family of curves for which a (N ) section exists that there exists a line bundle L on J X × S X × S X whose pull-back via ξ −1 ⊗N is equal to E = (D 12 ⊗ D 13 ) . def Now let us consider the “determinant of the higher direct image sheaves” (cf. [MB], §1, for an exposition of this notion) of L and E for the morphisms X × S X × S X → X × S X (N ) (N ) and J X × S X × S X → J X × S X given by forgetting the third factor. We denote the respective “determinants of the higher direct image sheaves” of L and E by L and E. Thus, (by the functoriality of forming the “determinant of the higher direct image sheaves”) L (N ) is a line bundle on J X × S X such that ξ ∗ (L) = E. Moreover, I claim that we can write E = D ⊗N ⊗ F ⊗m (for m ∈ Z) where F is a line bundle on X × S X obtained by pulling back some ω X/S from the ﬁrst factor of X × S X, and tensoring with the pull-back to X × S X of some line bundle M on S. Indeed, the “D ⊗N -term” arises from the fact that D 12 is the pull- back of D via the projection X × S X × S X → X × S X under consideration. Then the diﬀerence between the “determinants of the higher direct image sheaves” of the line −N bundles D 13 = O X× S X× S X (−N ·Δ 13 ) and O X× S × S X may be computed using the natural inclusion −N = O X× S X× S X (−N · Δ 13 ) ⊆ O X× S × S X D 13 By using the fact that the “determinant of the higher direct image sheaves” is multiplicative on exact sequences, we thus obtain that this diﬀerence is equal to some power of the pull- back to X × S X (via the projection to the ﬁrst factor) of O X× S X (−Δ)| Δ=X = ω X/S . On the other hand, the “determinant of the higher direct image sheaves” of the trivial line bundle on X × S X × S X is clearly the pull-back (to X × S X) of a line bundle M on S. This gives us a line bundle F of the form discussed above, hence completes the proof of the claim. By replacing L with L ⊗ O S M −1 , we may assume that M is trivial. Moreover, by (N ) Lemma 7.2 below, there exists a line bundle P on J X whose pull-back to X (via the (N ) natural morphism X → J X ) is some nonzero tensor power of ω X/S . Thus, replacing L by a tensor product of appropriate powers of L and (P| J (N ) × S X ) completes the proof of X the Lemma. 41 (N ) Lemma 7.2. There exists a line bundle P on J X tensor power of ω X/S . whose pull-back to X is a nonzero Proof. Let us ﬁrst consider the case N = g − 1. In this case, we take P to be the line (g−1) (g−1) bundle on J X deﬁned by the natural theta divisor on J X (i.e., the image of the (g−1) natural morphism from the (g − 1)-fold product of X to J X ). Then the fact that the ⊗ 1 g(g−1) 2 pull-back of P to X is equal to ω X/S is an immediate consequence of [MB], Corollary 2.5: in the notation of loc. cit., we are interested here in the case where one takes n = 0; S = X; and a : S → X to be the identity; then specializing the formula of loc. cit. to (a) the zero section of the Jacobian (note that the restriction of “U n ” in loc. cit. to the zero section of the Jacobian is trivial) proves the assertion concerning the pull-back of P to X. Next, let us consider the case where N is divisible by g − 1. Then the morphism (N ) (g−1) (N ) X → J X factors through J X → J X . Moreover, it follows from the basic theory of line bundles on abelian varieties (see, e.g., [AV], §23, the Corollary to Theorem 2 on p. 231) that a nonzero tensor power of the “P” considered in the previous paragraph descends (g−1) (N ) from J X to J X . This completes the proof in the case where N is divisible by g − 1. Finally, we consider the case of arbitrary positive N . In this case, we have a natural (N ) (N (g−1)) map J X → J X (multiplication by g − 1). But, by the preceding paragraph, we (N (g−1)) (N ) . Thus, by pulling this line bundle back to J X , already have a suitable “P” on J X (N ) we obtain a suitable “P” on J X . This completes the proof of the Lemma. def Next, we would like to consider a section α : Γ S = π 1 (S) → Π X of Π X → Γ S . By (N ) composing α with π 1 (−) applied to X → J X , we obtain a section α N J : Γ S → Π (N ) . J X (N ) Lemma 7.3. Suppose that α N J arises from a geometric section ∈ J X (S). Then there exists a line bundle on X of degree prime to p. Proof. By taking the ﬁbered product (over Γ S ) of α with the identity on Π X , we obtain a morphism α X : Π X → Π X× S X . Since S is the spectrum of the ﬁeld, the group cohomology of Π X× S X computes the p-adic étale cohomology of X × S X (cf. Lemma 0.4). Thus, we can form the arithmetic ﬁrst Chern class of the line bundle D of Lemma 7.1 (cf. def Deﬁnition 0.3): c 1 (D) ∈ H 2 (Π X× S X , Z p (1)). Let ζ = α ∗ X (c 1 (D)) ∈ H 2 (Π X , Z p (1)). On the other hand, by composing α X with π 1 (−) of the morphism ξ, we obtain a morphism . Moreover, by assumption, α N α N ξ : Π X → Π (N ) ξ arises from a geometric morphism J X × S X (N ) (N ) ∗ X → J X × S X. Thus, if L is a line bundle on J X × S X, we obtain that (α N ξ ) (c 1 (L)) ∈ H 2 (Π X , Z p (1)) can be written as c 1 (M), for some line bundle M on X. Now recall that (N ) by Lemma 7.1, there exists a line bundle L on J X × S X such that ξ ∗ L is a nonzero tensor power of D. Thus, putting everything together, it follows that some nonzero multiple of 42 the abstract cohomology class ζ ∈ H 2 (Π X , Z p (1)) can be written in the form c 1 (M) for some line bundle M on X, i.e., m · ζ = c 1 (M) (for some nonzero m ∈ Z). Next, let us recall that since ζ was constructed from D, the image of ζ under the “degree map” H 2 (Π X , Z p (1)) → H 2 (Δ X , Z p (1)) ∼ = Z p is equal to 1, so deg(M) = m. Thus, we can argue as follows (cf. [Mzk2], Lemma 6.1): Write m = a · p b , where a is an integer prime to p, and b is a nonnegative integer. Thus, c 1 (M) vanishes in H 2 (Π X , (Z/p b Z)(1)). But, by the Kummer exact sequence, this implies that there exists a line bundle P on X such that b P ⊗p ∼ = M. In particular, deg(P) = p −b · deg(M) = a, so the existence of P completes the proof of the Lemma. Now let us consider the following situation: Let K be a ﬁnite extension of Q p . Let X K → Spec(K) and Y K → Spec(K) be proper hyperbolic curves over K. Let U K be the generic point of X K . Moreover, let us assume that we have been given a continuous open homomorphism over Γ K θ : Π U K → Π Y K Now let S, η S be as at the beginning of Section 6. Write X η S for X K × K η S (and similarly, for U η S , Y η S ). Then by base-change (note that Π U η = Π U K × Γ K Γ η S ; Π Y η = Π Y K × Γ K S S Γ η S ), θ induces a continuous open homomorphism θ η S : Π U η → Π Y η S S Let Y η S → Y η S be a ﬁnite étale covering induced by some open subgroup of Π Y η that S surjects onto Γ η S . Note that θ η S allows us to deﬁne the pull-back of Y η S → Y η S to U η S . Let U η S → U η S be a connected component of this pull-back which is geometrically connected over η S . Note that U η S → U η S extends to a ﬁnite (possibly ramiﬁed) covering X η S → X η S . Thus, X η S is a proper hyperbolic curve over η S . Let us suppose that Y η S is irreducibly splittable (Deﬁnition 6.3). Then it follows immediately that X η S is also irreducibly splittable. (Indeed, the deﬁnition of “irreducibly splittable” only involves the base (i.e., S, S , etc.), plus ﬁnite étale coverings of the curve, i.e., it may be phrased entirely in terms of (the base plus) fundamental groups. Thus, the fact that X η S and Y η S are related by θ η S is enough to guarantee that “Y η S irreducibly splittable =⇒ X η S irreducibly splittable.”) Next, we would like to consider a section α U : Γ η S → Π U η S of Π U η → Γ η S . Composing α U with π 1 (−) of the natural morphism U η S → X η S gives rise S to a section α X : Γ η S → Π X η . Composing α U with the morphism θ η S : Π U η → Π Y η S S S induced by θ η S gives a section α Y : Γ η S → Π Y η . S 43 The following key result is the culmination of our eﬀorts in Sections 1 through 7: Proposition 7.4. Under the circumstances just described above, let us assume that α X is F I-geometric (Deﬁnition 6.4). Then it follows that Y η S admits a line bundle of degree prime to p. Proof. First observe that the morphism Δ U η → Δ Y η → H Y I η (where H Y I η is as in the S S S S I discussion preceding Deﬁnition 6.4) factors through H X . (Indeed, this follows by observ- η S ing, as in the proof of Lemma 5.2, that the inertia groups of Δ U η map to zero in H Y I η .) S S I Thus, we obtain a Γ η S -equivariant morphism H X → H Y I η . In particular, it follows by ηS S using Tate’s theorem (i.e., Theorem 4 of [Tate] – that morphisms between Tate modules of p-divisible groups induce morphisms between the p-divisible groups, hence morphisms FI 1 FI between the respective formal groups) that H f 1 (S K , H X ) maps to H f (S K , H Y ), for any ηS ηS ﬁnite étale covering η S → η S of η S as in Deﬁnition 6.2. Next, I claim that α Y is F I-geometric. Indeed, this follows from Proposition 5.4, plus the observation of the preceding paragraph. In words: This claim amounts to the claim (cf. Deﬁnition 6.4) that the “J -portion” of α Y (i.e., the result of composing α Y with (1) Π Y η → Π (1) , where J Y is the Picard scheme of line bundles of degree 1 on Y η S ) diﬀers J Y S from the “J -portion” of a constant (i.e., arising from a point deﬁned over a ﬁnite extension of K of the curve “Z” of the discussion of Section 6) geometric section of Π Y η by a class S in H f 1 (S K , H Y F η I ). On the other hand, since we are operating under the assumption that S α X is F I-geometric, we know that the J -portion of α X diﬀers from the J -portion of a FI constant geometric section of Π X η by a class in H f 1 (S K , H X ). Moreover, by Proposi- η S S tion 5.4, constant geometric sections of Π U η map to constant J -geometric (which is as S good as “geometric” for us, since we are only interested in “J -portions” here) sections of FI , H X Π Y η . Thus, since (by the observation of the preceding paragraph) H f 1 (S K ) maps η S S to H f 1 (S K , H Y F η I ), we conclude that α Y is F I-geometric, as desired. This completes the S proof of the claim. Now since Y η S is also assumed to be irreducibly splittable, it follows from Proposition 6.5 that α Y is, in fact, F -geometric. (N ) Now let us consider the morphism α N J : Γ η S → Π (N ) (where J Y J Y is the Picard scheme of line bundles of degree N on Y η S ), for some N > 0, induced by composing α Y with Π Y η → Π (N ) . Note that the F -geometricity of α Y is, by deﬁnition, a property J Y 1 concerning α J . By the deﬁnition of F -geometricity (Deﬁnition 6.4), it follows immediately (N ) that if N is large enough, then α N J arises from a geometric section J Y (S K ). Indeed, the S large N is to take care of the phrase “nonzero multiple of” in Deﬁnitions 6.2 and 6.4, plus the fact that a priori the geometric point whose existence is guaranteed by Deﬁnition 6.2 is only deﬁned over η S , so we may need to apply the norm map (for η S → η S ) to get a 44 geometric point over η S , which may cause N to increase again. Now, applying Lemma 7.3 completes the proof of the Proposition. Remark. Note that Sections 6 and 7 are made much more technically diﬃcult by the fact that we start with a morphism Π U K → Π Y K rather than a morphism Π X K → Π Y K . Indeed, the whole business of distinguishing “F -geometric” from “F I-geometric” arises because one does not have a proper theory of “p-divisible groups of inﬁnite rank,” which is the sort of object that one must deal with if one tries to work directly with H U (the abelianization of Δ U ) without passing to H U I . Similar technical problems (arising from the fact that H U is of inﬁnite rank) also are the reason behind the lengthiness of Sections 11 and 12. Section 8: From Line Bundles to Tame Points In this Section, which is something of an appendix to that portion of the paper con- stituted by Sections 1 through 7, we again apply some elementary algebraic geometry, this time to pass from line bundles to rational points of the curve deﬁned over tamely ramiﬁed extensions of the given ﬁeld. Thus, let M be a p-adic ﬁeld whose residue ﬁeld k M is iso- morphic to k((t)), where k is a ﬁnite ﬁeld, and t is an indeterminate. Let X M be a proper hyperbolic curve over M. Then we have the following result: Proposition 8.1. Suppose that X M admits a line bundle L of degree prime to p. Then there exists a tamely ramiﬁed extension M of M such that X M (M ) is nonempty. Proof. By replacing L by an appropriate (prime to p) tensor power of the original L, we may assume that L is very ample. Thus, by Bertini’s Theorem, it follows that there exists a divisor D ⊆ X M such that L ∼ = O X M (D), and D is étale over M. Let us write D = r D i i=1 where each D i = Spec(M i ). Thus, M i is a ﬁnite ﬁeld extension of M. Since deg(D) is prime to p, it follows that at least one of the M i (say, M 1 ) is such that [M 1 : M] is prime to p. Let M c be the Galois closure of M 1 over M. Then it follows from the elementary theory of p-adic ﬁelds (in particular, the fact that the wild inertia subgroup of Γ M is a normal pro-p subgroup of Γ M , hence contained in any Sylow-pro-p subgroup of Γ M – see, e.g., [Ser2], Chapter IV, §2) that M c is tamely ramiﬁed over M. Thus, in particular, M 1 is tamely ramiﬁed over M, and X M (M 1 ) is nonempty. This completes the proof of the Proposition. 45 Next, before continuing, we would like to discuss an auxiliary Lemma that will be useful when we apply Proposition 8.1. Let us suppose that there exists a p-adic ﬁeld L ⊆ M satisfying the following properties: (1) We have an inclusion O L ⊆ O M , relative to which m L · O M = m M . (2) If k L is the residue ﬁeld of L, then k ⊆ k L , and k L is a function ﬁeld in one variable over k. (3) The inclusion k L ⊆ k M is obtained by completing the function ﬁeld k L at one of its k-valued points. Let us suppose, moreover, that there exists a curve X L → Spec(L) such that X M = X L ⊗ L M. Then we have the following result: Lemma 8.2. Suppose that there exists a tamely ramiﬁed extension M of M of ramiﬁca- tion index e such that X M (M ) is nonempty. Then there exists a tamely ramiﬁed extension L of L of ramiﬁcation index e such that X L (L ) is nonempty. Proof. Since tamely ramiﬁed extensions of M descend to tamely ramiﬁed extensions of L, it is easy to see that without loss of generality, we may assume that e = 1, and M = M. Thus, we must show that X L admits a rational point over some unramiﬁed extension of L. Note that since O L is a discrete valuation ring, it is well-known that X L def admits a regular model X L → Spec(O L ). Let X M = X L ⊗ O L O M . Let us consider the morphism φ : Spec(O M ) → Spec(O L ). Since ﬁnitely generated algebras over a ﬁnite ﬁeld are excellent, it follows that k M is separable over k L (i.e., the morphism Spec(k M ) → Spec(k L ) is geometrically regular). In particular, it follows that φ is geometrically regular. Since the natural morphism X M → X L is obtained from φ by base-change, it thus follows that it, too, is geometrically regular. Thus, the regularity of X L implies that of X M . Now the fact that X M (M) is nonempty (by assumption) implies that there exist points in the special ﬁber of X M at which X M is smooth over O M . But this implies (by descent) that there exist points in the special ﬁber of X L at which X L is smooth over O L , which, in turn, implies that for some unramiﬁed extension L of L, X L (L ) is nonempty, as desired. Section 9: Convergence via p-adic Hodge Theory Whereas Sections 1 through 8 formed a unit devoted essentially to showing Propo- sition 7.4, Sections 9 and 10 form a new unit, devoted to showing, by means of p-adic Hodge theory, that certain types of sequences of points converge p-adically to a uniquely determined point deﬁned over a relatively small ﬁeld. Thus, in particular, we shall start with fresh hypotheses, as follows: 46 Let K be a p-adic local ﬁeld, with residue ﬁeld k. Let L be a p-adic ﬁeld containing K whose residue ﬁeld k L is a function ﬁeld in one variable over k. Let us also assume that k is algebraically closed in k L , and that m K · O L = m L . Let us denote by def Γ L/K = Ker(Γ L → Γ K ) = Γ L·K 1 the “geometric” fundamental group of L. Let H Ω L be the quotient of H (Γ L/K , O L (1)) by its def torsion submodule. Let X K → Spec(K) be a proper hyperbolic curve. Let X L = X K ⊗ K L. Thus, as usual, we have a group extension (obtained by pulling back 1 → Δ X → Π X K → Γ K → 1 via Γ L → Γ K ) 1 → Δ X → Π X L → Γ L → 1 Let us assume that we are given a continuous, group homomorphism α : Γ L → Π X K whose composite with the projection to Γ K is the natural morphism Γ L → Γ K . As usual, α deﬁnes a section α s : Γ L → Π X L of Π X L → Γ L . Let us also assume that α is nondegenerate in the sense that it satisﬁes the following group-theoretic condition: (∗) non The natural morphism H 1 (Δ X , Z p (1)) → H Ω L induced by α is nonzero. def Let L tm be a maximal tamely ramiﬁed extension of L. Let M = (L tm ) ∧ be its p-adic completion. Since L tm · K is unramiﬁed over L · K, and Galois cohomology “ignores unramiﬁed extensions,” we have that 1 (1)) = H 1 (Γ H 1 (Γ L/K , O L L·K , O L (1)) = H (Γ L tm ·K , O L (1)) In particular, since H Ω L is a quotient of this cohomology module, it follows from assumption non (∗) that α| Γ L tm ·K : Γ L tm ·K → Δ X is nontrivial. def For n ≥ 0, let I L n = Im(α s ) · (Δ X ) <n> ⊆ Π X L (cf. the discussion preceding Deﬁnition 0.2 for an explanation of the notation “< n >”). Let ψ n : X L n → X L be the corresponding ﬁnite étale covering. Let X L ∞ → X L be the inverse limit of the X L n . In this Section, we would like to prove the following assertion: (∗) con Suppose that we have a sequence of points { x n }, where x n ∈ def X L n (L tm ). Let x n = ψ n ( x n ) ∈ X L (L tm ). Then there exists a subse- quence of {x n } which converges p-adically in X L (M). 47 (Note that by choosing any proper model over O L of X L , we obtain a p-adic topology on X L (L tm ) which is easily seen to be independent of the model chosen.) By replacing K be a ﬁnite extension of K, we may make the following simplifying assumptions: (1) X K has stable reduction. We denote by X → Spec(O K ) the unique stable extension of X K over O K . We denote by J → Spec(O K ) the unique semi-abelian scheme over O K whose generic ﬁber is the Jacobian of X K . (2) X K (K) is nonempty. Note that replacing K by a ﬁnite extension of K does not aﬀect the validity of the con- vergence assertion (∗) con . Let us ﬁx an n ≥ 1. Let us consider the morphism x n : Spec(L tm ) → X K . This morphism induces a morphism β n : Γ L tm → Π X K → Π X K /(Δ X ) <n> which is well-deﬁned up to conjugation by an element of Δ X . Then it follows immediately from the deﬁnitions (and the fact that x n arose from an L tm -rational point of X L n ) that: This morphism β n is equal to the composite of α| Γ L tm : Γ L tm → Π X K with the natural projection Π X K → Π X K /(Δ X ) <n> . Thus, in particular, β n is independent of the choice of x n . Moreover, by (∗) non , it follows that by taking n to be suﬃciently large, we may assume that the restriction of β n to Γ L tm ·K (⊆ Γ L tm ) is nontrivial. This implies, in particular, that x n does not factor through any ﬁnite extension of K. Note that by properness, x n extends to a morphism ξ n : Spec(O L tm ) → X . Let K tm be the algebraic closure of K in L tm . Thus, K tm is a maximal tamely ramiﬁed extension of K. Let us denote by Ω O L tm /O K tm the O L tm -module of p-adically continuous diﬀerentials of O L tm over O K tm . Thus, Ω O L tm /O K tm is a free O L tm -module of rank one. Lemma 9.1. Ω O L tm /O K tm . By diﬀerentiating ξ n , we obtain a natural morphism dξ n : ξ n ∗ ω X /O K → Proof. Suppose that x n is deﬁned over some ﬁnite tamely ramiﬁed extension L of L. Suppose, moreover, that L contains a ﬁnite tamely ramiﬁed extension K of K such that m K · O L = m L . Then, if we equip Spec(O K ), Spec(O K ) and Spec(O L ) with the log structures deﬁned by the special points, and X with the log structure whose monoid is the sheaf of functions invertible on the generic ﬁber, we obtain a log morphism 48 Spec(O L ) log → X log compatible with ξ n and Spec(O K ) log → Spec(O K ) log . Moreover, since ω X /O K = Ω X log /Spec(O K ) log , and Ω O L tm /O K tm = O L tm ⊗ O L Ω Spec(O L ) log /Spec(O K ) log , we thus obtain dξ n by means of the functoriality of logarithmic diﬀerentials. The naturality of dξ n is easily checked by looking at the generic ﬁbers. def 0 Let Γ Ω X = H (X , ω X /O K ). Then we propose to prove in the present situation that (∗) ind There exists a number n 0 depending only on K such that for all n−n 0 n ≥ n 0 , the natural morphism Γ Ω Z induced X → Ω O L tm /O K tm ⊗ Z/p by dξ n is independent of x n (i.e., depends only on α). To keep the notation simple, let us note that by replacing K by a tamely ramiﬁed extension of K, and then replacing L by an unramiﬁed extension of L, we may assume that x n is, in fact, deﬁned over L. (Note that replacing L and K by extensions in this fashion will not aﬀect the validity of (∗) ind .) Thus, we shall regard x n and ξ n as morphisms Spec(L) → X K and Spec(O L ) → X , respectively. def Ω Let P = P(Γ Ω X ). (That is, P is the projective space over O K deﬁned by Γ X .) Then there is (by the deﬁnition of Γ Ω X ) a natural ﬁnite morphism λ K : X K → P K . Let N → Spec(O K ) be the Néron model of J X . Thus, J ⊆ N is an open subscheme of N . (1) Since X K (K) is nonempty, we can use a K-rational point of X K to identify J X with J X . Thus, we obtain a morphism X K → N K = J X . By the deﬁning property of the Néron model, and the fact that O L is geometrically regular over O K , it follows that composing this morphism with x n gives rise to a morphism ζ L : Spec(O L ) → N . Moreover, (after possibly replacing K by a ﬁnite unramiﬁed extension of K), we may assume that there exists some ζ K : Spec(O K ) → N such that ζ K and ζ L map the special points of Spec(O K ) and Spec(O L ), respectively, to the same geometric connected component of the special ﬁber (1) of N . Thus, by translating by ζ K , we may assume that our identiﬁcation of J X with J X is such that ζ L maps into J ⊆ N . We denote the resulting morphism by j n : Spec(O L ) → J . Now let us consider the Γ K -module H 1 (X K , Z p (1)) = H 1 (J K , Z p (1)). Here, the iden- tiﬁcation of H 1 (X K , Z p (1)) with H 1 (J K , Z p (1)) is the identiﬁcation induced by X K → (1) J X ∼ = J X . This identiﬁcation is the same as the standard identiﬁcation since translation by K-valued points of J induces the identity on H 1 (J K , Z p (1)). Let H cb ⊆ H 1 (J K , Z p (1)) be the maximal Z p -submodule such that Γ K acts on H cb (−1) through a ﬁnite, unram- iﬁed quotient. (Here, “cb” stands for “combinatorial.” Note that this inclusion H cb ⊆ F H 1 (J K , Z p (1)) is Cartier-dual to the quotient H X → H X /H X (in the notation of the dis- cussion following Deﬁnition 6.3 in Section 6).) By the theory of [FC], Chapter III, we know def that C X = H 1 (J K , Z p (1))/H cb may be (Γ K -equivariantly) identiﬁed with H 1 (G K , Z p (1)) for some semi-abelian scheme G → Spec(O K ) which can be represented as an extension 0 →T →G→A→ 0 49 of an abelian scheme A → Spec(O K ) by a torus T → Spec(O K ). Moreover, the p-adic completion G of G is equal to the p-adic completion J of J . Indeed, relative to the identiﬁcation G = J , the identiﬁcation C X = H 1 (G K , Z p (1)) is obtained as follows: Given a ﬁnite étale covering Q K → J K , we let Q be the normalization of J in Q K , so Q is → J = G ﬁnite over J . Thus, by p-adically completing, we obtain a formal morphism Q (whose relative diﬀerentials are annihilated by a power of p) which can be algebrized to some ﬁnite R → G (whose relative diﬀerentials are annihilated by a power of p). Thus, R K → G K is ﬁnite étale. In other words, the correspondence between coverings given by Q K → R K induces a morphism H 1 (J K , Z p (1)) → H 1 (G K , Z p (1)) whose kernel is H cb , and thus allows us to identify H 1 (G K , Z p (1)) with C X . Now by p-adically completing j n : Spec(O L ) → J , applying G = J , and then alge- brizing, we thus obtain a morphism g n : Spec(O L ) → G. Moreover, g n induces a morphism on cohomology groups H 1 (g n ) : H 1 (G K , Z p (1)) → H 1 (Γ L/K , Z p (1)) Thus, we have a commutative diagram of Γ K -modules: H 1 (J K , Z p (1)) ⏐ ⏐ H 1 (G K , Z p (1)) H 1 (j n ) −→ H 1 (g n ) −→ H 1 (Γ L/K , Z p (1)) ⏐ ⏐ id −→ n H Ω L ⊗ (Z/p Z) ⏐ ⏐ id H 1 (Γ L/K , Z p (1)) −→ n H Ω L ⊗ (Z/p Z) where the vertical morphism on the left is the natural projection discussed in the preceding paragraph, and the horizontal morphisms on the right are the natural ones. Recall that 1 H Ω L is, by deﬁnition, the quotient of H (Γ L/K , O L (1)) by its torsion submodule. Now, by sorting through the deﬁnitions (cf. the discussion preceding Lemma 9.1), it is clear that the composite morphism on the upper row is completely determined by α (i.e., is independent of the particular choice of x n ). Thus, it follows that the composite morphism on the bottom row gives rise to a morphism n : H 1 (1)) → H Ω ⊗ Z/p n Z (G K , O L K which is completely determined by α. Moreover, since H Ω L is p-adically separated (see Con- sequence (2) of [Falt1] below), it follows (from (∗) non ) that n is nonzero for n suﬃciently large. Now, note that G is a smooth O K -scheme with an obvious compactiﬁcation G such that G − G is a divisor with normal crossings. (Indeed, G is a product of G m -torsors over the O K -proper scheme A, so we simply compactify each of these G m -torsors to a P 1 -bundle.) In particular, it follows that we can apply the theory of [Falt1], in the case of good reduction. 50 Remark. The argument of [Falt1] in the bad reduction case is known to have gaps, which is one reason why we took pains to avoid applying [Falt1] to J K . Another reason is that in the bad reduction case, one only has “Q p -results,” not integral results (say, over Z/p n Z), as we need here. Let n 0 be an integer such that the p-adic valuation of the diﬀerent of K over Q p is ≤ def def the p-adic valuation of p n 0 . Let n 0 = n 0 + ord p (g!). Let n 0 = n 0 + 2. The consequences of the theory of [Falt1] that we use are as follows: ∧ (1) The Γ K -module H 1 (G K , K ) is Hodge-Tate. Let us denote by H Ω G the 1 quotient of H (G K , O K (1)) by the submodule that is contained in the ∧ portion of H 1 (G K , K (1)) of weight 1. Thus, H Ω G ⊗ Z p Q p is of weight inv 0. Let Ω G be the O K -module of invariant diﬀerentials (over O K ) on the group scheme G. Then there is a natural Γ K -equivariant morphism (Theorems 2.4 and 3.1 of II. of [Falt1]) n 0 n Ω inv · O K )/(p n 0 +n · O K )} → H Ω G ⊗ O K {(p G ⊗ Z/p Z with an inverse “up to a factor of p n 0 .” Here the point of the n 0 is to take care of the “g! · ρ” of [Falt1]: the 2 = 1 + 1 added to n 0 is to take 1 care of the p−1 ≤ 1 that always arises, plus the additional e−1 e < 1 that arise from the tame ramiﬁcation which we allow (see [Fo], Theorem 1 for more details on the computation of [Falt1]’s “ρ”). (2) H Ω L is p-adically separated. Moreover, there is a natural Γ K -equivariant morphism (Theorem 4.2 of I. of [Falt1]) n Ω O L /O K ⊗ O K {(p n 0 · O K )/(p n 0 +n · O K )} → H Ω L ⊗ Z/p Z with an inverse “up to a factor of p n 0 .” In particular, by consid- ering weights, it follows that n factors through the quotient H Ω G of 1 H (G K , O K (1)). (3) The morphisms (and inverses up to a factor) of (1) and (2) are com- patible with each other, n , and the natural morphism Ω inv G → Ω O L /O K induced by diﬀerentiating g n . That is to say, we have a commutative diagram: n 0 · O K )/(p n 0 +n · O K )} Ω inv G ⊗ O K {(p ⏐ ⏐ −→ n H Ω G ⊗ Z/p Z ⏐ ⏐ Ω O L /O K ⊗ O K {(p n 0 · O K )/(p n 0 +n · O K )} −→ n H Ω L ⊗ Z/p Z 51 (where the horizontal morphisms are the morphisms of (1) and (2), re- spectively, the vertical morphism on the left is induced by diﬀerentiating g n , and the vertical morphism on the right is induced by n ). These assertions allow us to immediately conclude (∗) ind , as follows: Since n is determined by α, it thus follows that the morphism (induced by diﬀerentiating g n ) Ω inv G → Ω O L /O K ⊗ (Z/p n−n 0 Z) is determined by α. On the other hand, Ω inv can naturally be identiﬁed with G inv inv Ω Ω J . Moreover, Ω J can naturally be identiﬁed with Γ X . This completes the proof of (∗) ind . Lemma 9.2. n is nonzero. Ω n For n suﬃciently large, the morphism H Ω G → H L ⊗ (Z/p Z) induced by Proof. This follows formally from (∗) non . Now we are ready to tackle the convergence assertion (∗) con . Thus, we go back to working over L tm and K tm . In particular, by composing x n with λ K and applying the valuative criterion for properness, we obtain a morphism λ n : Spec(O L tm ) → P . Moreover, this O L tm -valued point λ n of the projective space P is precisely the point deﬁned by the → Ω O L /O K ⊗ O L O L tm = Ω O L tm /O K tm (obtained by diﬀerentiating g n ). morphism Ω inv G (Note that here we use the fact remarked above that Ω O L tm /O K tm is a free O L tm -module of rank one.) Thus, (∗) ind and Lemma 9.2 imply that the points λ n of P (O L tm ) converge L tm ) = P (M). On the other hand, by “Krasner’s Lemma” p-adically to a point λ ∞ ∈ P ( O (see, e.g., [Kobl], p. 70), since λ K : X K → P K is ﬁnite, it follows that some subsequence {x n i } of {x n } converges to a point x ∞ ∈ X K (M). This proves the assertion (∗) con , as desired. In fact, we can say more. For m ≥ 1, let x m n+m in X L m (L tm ) under n be the image of x n+m the natural morphism X L → X L m . Thus, we obtain a sequence { x m n } (in the index n) of m tm m points of X L (L ). Since X L → X L is ﬁnite, then by appying “Krasner’s Lemma” again, m see that a subsequence of the sequence { x m n } converges p-adically in X L (M). Moreover, by the process of “Cantor diagonalization” (of elementary analysis), we thus see that we have proven the following key result (which is the main result of this Section): Lemma 9.3. We assume notation as in the ﬁrst paragraph of this Section. Suppose that n ∈ X L n (L tm ). Then there exists a subsequence we have a sequence of points { x n }, where x { x n i } of this sequence with the following property: For each m ≥ 0, the sequence obtained m by projecting those x n i with n i ≥ m to X L m (M) converges to some x m ∞ ∈ X L (M). In particular, X L ∞ (M) is nonempty. m Proof. The last assertion is proven by noting that the x m ∞ ∈ X L (M) form a compatible ∞ system, hence deﬁne a point of x ∞ ∞ ∈ X L (M). 52 Finally, before proceeding, we make the following technically trivial, but important observation: Since the arithmetic fundamental group of X L ∞ is just Γ L (i.e., Im(α s )), it follows that the morphism Γ M → Π X K ∞ induced on fundamental groups by the image in X K (M) of any point (e.g., x ∞ ∞ ) of X L (M) is none other than the restriction of α : Γ L → Π X K to Γ M . Section 10: Uniqueness and Rationality of the Limit Point We continue with the notation of the preceding Section. In this Section, we would ∞ like to show that the point x ∞ ∞ ∈ X L (M) constructed in Lemma 9.3 is the unique element of X L ∞ (M), and that it in fact descends to a point of X L ∞ (L). We remark that much of the material of this Section is not absolutely logically necessary for the proof of the main results of this paper, but is included partly for aesthetic reasons, and partly because its inclusion (in the opinion of the author) makes the proof of the main results of the paper more transparent. First, let us make the trivial observation that any point in X L ∞ (M) arises as a x ∞ ∞ (as n tm in Lemma 9.3) for some sequence of x n ∈ X L (L ). (Indeed, this follows by using the fact that M = (L tm ) ∧ to approximate the images of the given point in the various X L n (M)’s.) Thus, without loss of generality, we can apply the results of the discussion of Section 9 to our analysis of an arbitrary point of X L ∞ (M). Now recall the limit point λ ∞ ∈ P (M) of the discussion following Lemma 9.2. This limit point is clearly independent of the choice of the particular sequence { x n } under ∧ consideration – i.e., it depends only on α. More precisely, λ ∞ ∈ P (M) ⊆ P (L ) is the point deﬁned by the surjection ∧ inv Γ Ω X ⊗ O K L = Ω G ⊗ O K L ∧ ∧ Ω = H Ω G ⊗ O L → H L ⊗ Z p Q p = Ω O L /O K ⊗ O L L ∧ K induced by α (cf. the discussion preceding Lemma 9.2). Note that Γ L acts naturally on all of these modules, and that this surjection is Γ L -equivariant. Indeed, this follows by transport of structure from the fact that α is deﬁned on Γ L , not just on Γ L tm ·K . Thus, we conclude the following ∞ Lemma 10.1. The image λ ∞ ∈ P (M) of x ∞ ∞ ∈ X L (M) under the natural morphism X L ∞ → X K → P K is independent of x ∞ ∞ and, moreover, λ ∞ is deﬁned over L. 53 The next step is to observe that this argument can be modiﬁed so as to show that n the image of x ∞ ∞ in the projective space deﬁned by the diﬀerentials on X L (for n ≥ 1) is independent of x ∞ ∞ and deﬁned over L. To see this, we need to introduce some new objects. First, let us ﬁx n. Now observe that (after possibly enlarging K) there exists a ﬁnite étale n → Spec(K) of X K with stable reduction Z n → Spec(O K ), together with a covering Z K of L such that ﬁnite Galois extension L n ∼ X n = X L n ⊗ L L ⊗ K L = Z n = Z K L L tm ), where L tm is the composite Now let m ≥ n, and consider x m ∈ X L m (L tm ) ⊆ X L m ( L (as extensions of L). (Note that L tm is then a maximal tamely ramiﬁed of L tm and L n Thus, by projecting x , we obtain a morphism extension of L.) m to a point of Z K n z m : Spec(O L tm ) → Z n Let us assume that we have chosen K large enough so that Z K (K) is nonempty, so that (1) n n (i.e., the Jacobian of Z we can identify J Z K n . Let N Z n → Spec(O K ) be the K ) with J Z K K n Néron model of Z K over O K . More generally, in the following discussion we shall denote by N (−) the Néron model of any proper hyperbolic curve “(−)” over a discretely valued ﬁeld. In fact, we shall even use this notation for proper hyperbolic curves over inductive limits of discretely valued ﬁelds in which case this notation is to be taken to mean the corresponding inductive limit of the Néron models. Also, we shall denote by Comp(N (−) ) the (inductive limit of) ﬁnite abelian (étale) group scheme(s) of connected components of the special ﬁber of the Néron model N (−) . The main technical diﬃculty that we must overcome in order to apply the argument n of Section 9 (for X K ) to the curves Z K is the following: The points that we are interested (1) ∼ n tm n in are the points z m of Z K ( L ) = Z (O L n = J Z n , tm ). These points deﬁne points of J Z K K hence points N n z m : Spec(O L tm ) → N Z L tm Then, in order to apply the argument of Section 9, we must show that: (After replacing K by a ﬁnite extension of K which is independent of N m), z m maps the special point of Spec(O L tm ) to the same component tm n of Comp(N Z ) as some K -valued point of N Z n . L tm L tm 54 (see Lemma 10.2 below for an alternate formulation.) To show this, we reason as follows: in L tm . Thus, L tm is a totally unr for the maximal unramiﬁed extension of L Let us write L tm is a union of ﬁnite Galois extensions unr . In particular, L tamely ramiﬁed extension of L unr which are of degree (over L unr ) prime to p. In particular, it follows that the cokernel of L of the inclusion Comp(N Z n ) → Comp(N Z n ) L unr L tm is annihilated by integers prime to p. (Indeed, this follows by applying the “trace map” tm over L unr ) to the second “Comp,” and observing that (since (for subextensions of L tm / L unr ) clearly acts trivially on Comp(N Z n )) this trace map is just multiplication Gal( L L tm unr is an unramiﬁed extension of by some integer prime to p.) On the other hand, since L it follows (essentially from the deﬁnition of the Néron model) that N Z n = N Z n . In L, L unr L particular, since L is a discretely valued ﬁeld, it follows that Comp(N Z n ) is ﬁnite. In other L words, we conclude that: the p-torsion of Comp(N Z n ) is annihilated by some (ﬁnite) L tm power of p. We are now ready to prove the following Lemma: Lemma 10.2. After possibly enlarging K by a larger ﬁeld that is independent of m, we J ∼ tm ) → J (1) may arrange that the morphism z m : Spec( L n = J Z n induced by z m extends to a Z K K n ) → N , for all m ≥ n. morphism Spec(O L Z tm tm K Proof. Indeed, without enlarging K, it follows from our observation above concerning the J n for some p-torsion of Comp(N Z n ) that a · z m extends to (z N ) m : Spec(O L tm ) → N Z K L tm positive integer a whose order at the prime p is independent of m. Now let us assume n annihilated (by enlarging K – independently of m) that all of the p-torsion points of J Z K n by a are rational over K. Since all prime-to-p torsion points of J Z K are rational over K tm , it thus follows that the morphism “multiplication by a” on N Z n tm is ﬁnite over a K n ⊗ O tm ⊆ N n O . Since (z N ) m neighborhood of the image of (z N ) m ⊗ O K O K tm in N Z K K K Z K tm J J was constructed by multiplying z m by a, it thus follows that z m extends to a morphism Spec(O L tm ) → N Z n tm , as desired. K Now that we have Lemma 10.2, we can use the extended morphism of Lemma 10.2 to construct the analogue of the morphism “g n ” of Section 9 (over some tamely ramiﬁed extension of K). Then the rest of the argument of Section 9 goes through without diﬃculty. More precisely: def def 0 n n n Let Γ Ω = P(Γ Ω Z n = H (Z , ω Z n /O K ). Let P Z n ). (That is, P is the projective space Ω Ω over O K deﬁned by Γ Z n .) Then there is (by the deﬁnition of Γ Z n ) a natural ﬁnite morphism n n n λ nK : Z K → P K . Let λ nm : Spec(O L be the morphism obtained by composing z m tm ) → P n with λ K . Then, just as in Section 9, one sees via p-adic Hodge theory that as m → ∞ (and 55 tm ) = P n (( L tm ) ∧ ). Moreover, n remains ﬁxed), the λ nm converge to a point in λ n ∞ ∈ P n ( O L x m }. Finally, just as we saw for Lemma 10.1, λ n ∞ is independent of the original sequence { by transport of structure and the fact that α (and, here, X L n ) are deﬁned over L, it follows n tm ) ∧ ), where P X n is the projective space over that if we regard λ n ∞ as a point of P X (( L L associated to H 0 (X L n , ω X L n /L ) n (for the original L), then λ n ∞ ∈ P X (L). Thus, we see that we obtain the following analogue of Lemma 10.1: ∞ ∞ Lemma 10.3. The image λ n ∞ ∈ P X (M) of x ∞ ∈ X L (M) under the natural morphism n n X L ∞ → X L n → P X is independent of x ∞ ∞ and, moreover, λ ∞ is deﬁned over L. n Now let us recall some basic facts on hyperelliptic curves: Lemma 10.4. Let Q be a proper hyperbolic curve over an algebraically closed ﬁeld Ω of characteristic zero. Then: (1) If Q is hyperelliptic (i.e., admits a “g 2 1 ,” or linear system of dimension 1 and degree 2), then the g 2 1 is unique. (2) If Q is non-hyperelliptic, then the canonical morphism from Q into the projective space associated to H 0 (Q, ω Q/Ω ) is an embedding. (3) Suppose that W → Q is a ﬁnite étale covering, where W is connected. Then if Q is non-hyperelliptic, so is W . (4) Suppose that W → Q (where W is connected) is a cyclic étale covering of degree m > 2. Then W is non-hyperelliptic. Thus, in particular, none of the X L n is hyperelliptic, for n ≥ 2. Proof. For proofs of assertions (1) and (2), we refer to [Harts], Chapter IV, §5, Propo- sitions 5.2 and 5.3. As for (3), the push-forward of a g 2 1 on W via W → Q is a g 2 1 on Q, so (3) follows immediately. Now let us consider assertion (4). Let σ be a generator of the Galois group of W over Q. If W admits a g 2 1 , it is unique, hence stabilized by σ. But this means that there exists some rational function f W on W in this g 2 1 which satisﬁes σ −1 (f W ) = T (f W ), where T is some linear fractional transformation with coeﬃcients in Ω. Note that since σ has ﬁnite order, so does T . Thus, if we diagonalize T (by choosing a diﬀerent f W ), we may assume that σ −1 (f W ) = λ · f W , for some nonzero λ ∈ Ω. (Note that T cannot be parabolic (i.e., a transformation of the form f W → f W + λ, for some m nonzero λ ∈ Ω) since T is of ﬁnite order and Ω is of characteristic zero.) Thus, f W (which 56 is a constant multiple of the norm (relative to W → Q) of f W ) deﬁnes a rational function f Q on Q. Now f Q is contained in some g 2 1 on Q, so its zeroes have order equal to 1 or 2. On the other hand, f Q has an m th root in the function ﬁeld of W , so it follows from the assumption that m > 2 that the covering W → Q must be ramiﬁed at the zeroes of f Q , which is absurd. This contradiction completes the proof. We summarize our eﬀorts in Sections 9 and 10 as follows: Corollary 10.5. Let M be the p-adic completion of a maximal tamely ramiﬁed extension L tm of L. Suppose that we are given a nondegenerate α : Γ L → Π X K , which thus gives rise to X L n , X L ∞ . Suppose that for each n ≥ 0, X L n (L tm ) is nonempty. Then it follows that the set X L ∞ (M) consists of precisely one point, which is, in fact, contained in X L ∞ (L) ⊆ X L ∞ (M). In particular, there exists a unique L-valued point of Spec(L) → X K whose induced morphism on fundamental groups (for an appropriate choice of base-points) is the mor- phism α : Γ L → Π X K . ∞ Proof. Indeed, if x ∞ if n ≥ 2, the image of x ∞ ∞ ∈ X L (M), then by Lemma 10.4, ∞ in n n n ∞ X X L (M) is determined by the image λ ∞ of x ∞ in P (M). But, by Lemma 10.3, λ n ∞ is ∞ n independent of x ∞ and deﬁned over L. Thus, it follows that the image of x ∞ ∞ in X L (M) ∞ (for n ≥ 2) is independent of x ∞ ∞ and deﬁned over L. But, by the deﬁnition of X L , this ∞ ∞ means that x ∞ itself is “independent of x ∞ ” and deﬁned over L. (In other words, X L ∞ (M) consists of precisely one point, which is, in fact, deﬁned over L.) The last sentence is a formal consequence of the rest of the Corollary. This completes the proof. Section 11: Hodge-Tate Representations of Inﬁnite Rank Let K be a ﬁnite extension of Q p . Let X K → Spec(K) be a proper hyperbolic curve over K. Let U K be its generic point. Let H U be the abelianization of Δ U . Thus, H U is a Z p -ﬂat topological Γ K -module of inﬁnite rank. Note that we have a natural surjection I I H U → H X → H X (cf. the discussion following Deﬁnition 6.3 for the deﬁnition of H X ). M Let H U ⊆ H U be the kernel of this surjection. Note that the inertia groups (deﬁned by closed points of X K ) of Δ U all map into H U M . Let H U P ⊆ H U M denote the closure of the image of all these inertia groups. (Here the “P ” stands for “points.”) Thus, H U /H U P may be identiﬁed with H X , hence is of ﬁnite rank over Z p . In this Section, we would like to analyze H U , and, in particular, H U P in greater detail. For simplicity, let us assume (by replacing K by a ﬁnite extension of K) that X K has stable reduction over O K , and that there exists a K-valued point x ∈ X K (K). Let ℘ ∈ X K be a closed point of X K . Then there are ﬁnitely many points y 1 , . . . , y r ∈ 57 X K (K) = Hom(Spec(K), X K ) that map to ℘. For i = 1, . . . , r, let I y i ⊆ Δ U denote “the” inertia group (well-deﬁned up to conjugation by an element of Δ U ) associated to y i . Thus, projection to the quotient Δ U → H U yields a map def I ℘ = r I y i → H U P i=1 Note, moreover, that I ℘ has a natural structure of Γ K -module (given, for instance, by conjugating by the image of a section Γ K → Π U K (cf. Deﬁnition 5.1) induced by x ∈ X K (K) – note, however, that the Γ K -action is independent of the choice of x). Relative to this Γ K -action on I ℘ , the above morphism is Γ K -equivariant. Moreover, by letting ℘ range over all closed points of X K other than that deﬁned by x, we obtain a continuous Γ K -equivariant morphism def I ℘ → H U P Ξ : M x = ℘ = x In fact, Lemma 11.1. The morphism Ξ is an isomorphism. Proof. Note that Π U K is equal to the inverse limit of Π V K , where the limit is taken over all open subsets V K ⊆ X K − {x}. For such a V K , we can deﬁne H V P just as we deﬁned H U P , and it is well-known that H V P is the direct sum of the I ℘ , where the sum is taken over all ℘ ∈ X K − {x} − V K – indeed, this follows from the well-known (from elementary algebraic topology) structure of the homology group of a Riemann surface obtained by removing a ﬁnite number of points from a compact Riemann surface. Passing to the limit proves the Lemma. Next, we would like to deﬁne a quotient I ℘ → I ℘ T as follows: Note that if K ℘ is the residue ﬁeld of X K at ℘, then the Γ K -module I ℘ (−1) (where the “(−1)” is a Tate twist) may be naturally identiﬁed with Z p [Hom K (K ℘ , K)]. (Here, Z p [a set] denotes the free Z p - module generated by the elements of the set.) Thus, there exists a unique nonzero quotient I ℘ (−1) → Q ℘ stabilized by Γ K such that Γ K acts trivially on Q ℘ . Moreover, Q ℘ is a free Z p -module of rank one. Let I ℘ → I ℘ T be the quotient obtained by tensoring I ℘ (−1) → Q ℘ with Z p (1). Moreover, by taking the product of these quotients, we obtain a quotient def M x → M x T = I ℘ T ℘ = x Let H U P → H U T be the quotient corresponding to this quotient under the isomorphism Ξ. Thus, the Γ K -action on H U T (−1) is trivial. Moreover, H U P → H U T has the following universal property: 58 Lemma 11.2. Let N be a Z p -ﬂat topological Γ K -module such that Γ K acts trivially on N (−1). Then any continuous Γ K -morphism H U P → N factors through H U T . Proof. This follows immediately from the construction of H U T . T Now let us deﬁne the subquotient H X of H U as follows: First, let H U F ⊆ H U denote the F inverse image of H X ⊆ H X under the projection H U → H X (cf. the discussion following F T Deﬁnition 6.3 for the deﬁnition of H X ). Now let H X be the quotient of H U F by the kernel P T of H U → H U . Thus, we have an exact sequence of topological Γ K -modules T F 0 → H U T → H X → H X → 0 Let us take the continuous dual Hom cont Z p (−, Z p ) of this exact sequence. This gives us a new exact sequence of topological Γ K -modules F T → C X → C U T → 0 0 → C X (Here, one may think of the “H’s” as standing for “homology,” and the “C’s” as standing for “cohomology.”) In particular, we have C U T = ∧ Z ℘ T ℘ = x where Z ℘ T is deﬁned to be the dual of I ℘ T . If we pull-back this last exact sequence by Z ℘ T ⊆ C U T , we obtain exact sequences of topological Γ K -modules F → C ℘ T → Z ℘ T → 0 0 → C X T is obtained by all of which are of ﬁnite rank over Z p . Moreover, it is clear that that C X summing these extensions (over ℘) and then p-adically completing. T We would like to show that C X is “Hodge-Tate”. Unfortunately, typically the theory of Hodge-Tate Galois representations only goes through for modules of ﬁnite rank over Z p . On the other hand, C ℘ T is of ﬁnite rank over Z p . Thus, our approach in the following will T be to use the C ℘ T to show that C X is, in some sense, like a Hodge-Tate representation. First, let X ℘ be the singular curve obtained from X K by considering the subsheaf of O X K of functions f such that f(x) = f(℘). Let J ℘ be the generalized Jacobian associated to X ℘ (see, e.g., [Ser1], Chapitre I, §1). Thus, J ℘ is an extension of the abelian variety J X by some torus. Moreover, the p-adic Tate module T (J ℘ ) of J ℘ ﬁts into an exact sequence of Γ K -modules 59 0 → I ℘ → T (J ℘ ) → H X → 0 In particular, one sees immediately that there exists a quotient J ℘ → J ℘ whose p-adic Tate module ﬁts into the exact sequence 0 → I ℘ T → T p (J ℘ ) → H X → 0 obtained by pushing forward the preceding exact sequence by I ℘ → I ℘ T . Moreover, C ℘ T F may be identiﬁed with the dual of the pull-back of this exact sequence by H X → H X . T ∨ 1 Alternatively, C ℘ may be identiﬁed with a certain quotient T p (J ℘ ) = H (J ℘ , Z p ) → C ℘ T F (whose kernel is the dual of H X /H X ). Note, in particular, that J ℘ is an extension of J X by the trivial torus G m . Thus, J ℘ extends to a semi-abelian variety over O K (cf. the paragraph following Theorem 2.6 of Chapter I of [FC]). Next, let us recall (from the theory of [FC], Chapters I through III) that J X may be constructed as a certain rigid analytic quotient of a semi-abelian variety J X which has good reduction (i.e., the dimension of the torus part is constant) over O K by some discrete group. The kernel of the induced pull-back map on cohomology H 1 (J X , Z p ) → H 1 ( J X , Z p ) F . If we pull-back the extension J ℘ → J X from J X to J X , is precisely the dual of H X /H X we then obtain an extension J ℘ of J X by the trivial torus of dimension one. Moreover, J ℘ has good reduction over O K . Finally, the quotient H 1 (J ℘ , Z p ) → H 1 ( J ℘ , Z p ) is precisely the quotient H 1 (J ℘ , Z p ) → C ℘ T referred to in the preceding paragraph. Now, just as in Section 9, we can apply the theory of [Falt1] (in the case of good def reduction) to J ℘ . Let F 0 = H 1 (X, O X ), where X → Spec(O K ) is the stable extension of X K over O K . Let F ℘ be the space of invariant diﬀerentials on J ℘ , where J ℘ → Spec(O K ) is the extension of J ℘ to a semi-abelian variety over O K . Thus, F 0 and F ℘ are both free of ﬁnite rank over O K . Then it follows from the theory of [Falt1] (speciﬁcally, Theorems 2.4 and 3.1 of II. of [Falt1]) that -linear morphism (∗) ℘ There exists a natural Γ K -equivariant, O K → (F 0 ⊗ O O ) ⊕ (F ℘ ⊗ O O (−1)) C ℘ T ⊗ Z p O K K K K K which has an inverse up to p n , where n is a nonnegative integer that is independent of ℘. Note that the fact that n is independent of ℘ is of profound importance in what follows. Moreover, if, instead of J ℘ , we had worked with J ℘ (which, in general, has “bad (though stable) reduction”), or J ℘ (which has only potentially stable reduction – i.e., for general ℘, one needs to enlarge K in order to get stable reduction), we would have been unable to obtain this crucial “independence of ℘” from the theory of [Falt1]. 60 Now observe that F ℘ ﬁts into a natural exact sequence F → F ℘ → Z ℘ → 0 0 → F X def F = H 0 (X, ω X/O K ), and Z ℘ is deﬁned so as to make the above sequence exact. where F X T Now let F X be the sum of these extensions over all ℘; let F U T be the direct sum of the Z ℘ over all ℘. Thus, we have an exact sequence of ﬂat O K -modules F T → F X → F U T → 0 0 → F X Thus, by taking the p-adic completion of the sum of the morphisms in (∗) ℘ , and using the facts that: (i) the “n” of (∗) ℘ is independent of ℘; T T (ii) both C X and F X are sums of extensions which are of ﬁnite rank; we obtain the following result: -linear morphism of exact sequences of Proposition 11.3. We have a natural, O K topological Γ K -modules 0 −→ F ⊗ Z p O C X K ⏐ ⏐ −→ T ⊗ Z p O C X K ⏐ ⏐ 0 −→ F (−1)) ⊕ (F 0 ⊗ ) O K O (F X ⊗ O K O K K −→ T (−1)) ⊕ (F 0 ⊗ ) O K O ⊗ O K O (F X K K −→ Z p O C U T ⊗ K ⏐ ⏐ −→ 0 −→ (−1)) O K O (F U T ⊗ K −→ 0 which become isomorphisms when tensored with Q p . T Next, we would like to relate F X to a certain space of diﬀerentials on X, as follows: def Let E be a ﬁnite set of closed points of X K . Let S = Spec(O K ), and let us endow S with the log structure deﬁned by the monoid O S − {0}; denote the resulting log scheme by S log . Let X log [E] → S log be the pointed stable log-curve (see, e.g., [Mzk3], Section 3, for more details on this terminology) extending X K ⊗ K K, equipped with all the points 61 def of X K (K) that map to primes in E as marked points. Let L[E] = ω X log [E]/S log . Then if E ⊆ E , we get a natural morphism X log [E ] → X log [E], which lies under a morphism of sheaves L[E] → L[E ]. Thus, by taking the projective limit of the (X log [E], L[E]) as E ranges over all ﬁnite sets of closed points of X K , we obtain (X log [∞], L[∞]). Moreover, note that H 0 (X[∞], L[∞]) is a ﬂat, p-adically separated O K -module. If E contains x and ℘, then it is easy to see that we get a natural inclusion F ℘ → H (X[E], L[E]), which extends to a morphism F ℘ ⊗ O K O K → H 0 (X[E], L[E]) whose cokernel is torsion free. Taking the limit with respect to E, we thus obtain a map F ℘ ⊗ O K O K → H 0 (X[∞], L[∞]). In fact, it is not diﬃcult to see that, by summing over the F ℘ for all ℘ = x, we get a morphism 0 def T F X ⊗ O K O K → F ∞ = H 0 (X[∞], L[∞]) whose cokernel is torsion free. Moreover, this morphism remains injective after p-adic completion. Thus, we see from Proposition 11.3 that we get a morphism T → F ∧ (−1) ⊗ Z p O Φ : C X ∞ K (where the “∧” denotes p-adic completion). Moreover, the kernel of Φ is of weight zero. def Finally, if N is a positive integer, let G ∞ [N ] = H 0 (X[∞], L[∞] ⊗N ). Thus, G ∞ [N ] is a free O K -module of inﬁnite rank (such that G ∞ [1] = F ∞ ), and we have a natural multiplication morphism ⊗ N O K F ∞ → G ∞ [N ] Section 12: The Preservation of Relations The purpose of this Section is to show how the material of Section 11 will be applied in Section 13 to show the “preservation of relations.” We maintain the notation of Section 11. Moreover, let us assume that we have been given a continuous open homomorphism over Γ K θ : Π U K → Π Y K (where Y K → Spec(K) is a proper hyperbolic curve over K). For simplicity, let us assume (by enlarging K) that Y K extends to a stable curve Y → Spec(O K ) over O K , and that the Γ K -action on H Y M (−1) (cf. the discussion following Deﬁnition 6.3 for an explanation of the notation “H M ”) is trivial. Next, observe that θ induces a continuous morphism of Γ K -modules H U → H Y whose cokernel is torsion. Note that this morphism induces 62 a map H U F → H Y F (cf. the discussions following Deﬁnition 6.3 and Lemma 11.2 for an explanation of the notation “H F ”) whose cokernel is torsion. Moreover, the restriction of this morphism to H U P (cf. the discussion at the beginning of Section 11 for an explanation of the notation “H U P ”) maps into H Y M , hence (by Lemma 11.2 and the fact that Γ K acts T trivially on H Y M (−1)), we get a morphism H X → H Y F whose cokernel is torsion. Taking T the dual of this morphism yields an injection C Y F → C X (where C Y F is the dual of H Y F ). Topologically tensoring with O K , dividing out by the “weight zero part,” and applying the morphism Φ considered at the end of Section 11, we thus obtain (after further tensoring with Z p (1)) an injection → F ∧ H 0 (Y, ω Y /O K ) ⊗ O K O ∞ K In this Section, we shall be concerned with the issue of whether or not this morphism “preserves relations.” By this, we mean the following: Let N be a positive integer. By taking the N th tensor power of this morphism, and then composing with the completion of the multiplication morphism ⊗ N F ∞ → G ∞ [N ] considered at the end of Section 11, we obtain a morphism ) → (G ∞ [N ]) ∧ Ψ N : ⊗ N (H 0 (Y, ω Y /O K ) ⊗ O K O K We would like to know whether or not Ψ N annihilates the elements of R N = Ker(⊗ N (H 0 (Y, ω Y /O K )) → H 0 (Y, ω Y ⊗N /O K )) def i.e., the “relations.” In the following, we would like to state a certain assertion that will be proven in the next Section, and explain why this assertion implies that Ψ N “preserves relations.” First, let us introduce the notation necessary to state the assertion. Suppose that E is a ﬁnite set of points of X K (K). Then the pointed stable curve X log [E] → Spec(O K ) introduced at the end of Section 11 is deﬁned over K, i.e., we have X log [E] O K → Spec(O K ). Let p ∈ X[E] O K be a generic point of the special ﬁber of X[E] O K . Let Spf(O L ) be the completion of the localization of X[E] O K at p. Thus, L (the quotient ﬁeld of O L ) is a p-adic ﬁeld whose residue ﬁeld is k(p) (i.e., the residue ﬁeld of X[E] O K at the prime p), and we have a canonical L-valued point ξ L ∈ X K (L). Let Ω L/K denote the module of p-adically continuous diﬀerentials of L over K. Thus, Ω L/K is an L-vector space of dimension one. Moreover, note that we have a natural restriction morphism (induced by ξ L ) ∧ K K → Ω L/K ⊗ F ∞ Thus, if we compose this morphism with the morphism → F ∧ H 0 (Y, ω Y /O K ) ⊗ O K O ∞ K 63 considered above, we get a morphism → Ω L/K ⊗ K K κ L : H 0 (Y, ω Y /O K ) ⊗ O K O K Next, let us observe that the coverings of X K deﬁned by subgroups of Π U K are étale at the point ξ L . Thus, it follows that ξ L induces a natural morphism α L U : Γ L → Π U K (well-deﬁned up to conjugation by an element of Δ U ) whose composite with Π U K → Γ K L is the natural morphism Γ L → Γ K . Let α L Y : Γ L → Π Y K be the composite of α U with θ. Then the assertion that will be proven in Section 13 is the following: (∗) L−pt Suppose that p satisﬁes the condition that κ L is not identically zero. Then it follows that there exists a point (ξ ) L ∈ Y K (L) such that L α L Y arises from (ξ ) . L Note in particular, that if α L Y arises from (ξ ) , then it follows immediately that the group- theoretically constructed morphism κ L coincides (cf. the application of the theory of [Falt1] discussed in Section 9) with the natural restriction map on diﬀerentials induced by (ξ ) L . Thus, if ρ ∈ R N is a relation, then it follows that the restriction of Ψ N (ρ) ∈ (G ∞ [N ]) ∧ to Ω ⊗N L/K ⊗ K K is zero. This argument already leads one to believe that there should be some sort of connection between (∗) L−pt and the “preservation of relations.” In fact, we have the following: Proposition 12.1. Suppose that (∗) L−pt always holds (i.e., for all data of the sort discussed above). Then Ψ N (R N ) = 0, for all positive integers N . def Proof. Suppose that ρ ∈ R N is such that ψ = Ψ N (ρ) = 0. By dividing ψ by a suitable element of O K , we obtain an element ψ ∈ (G ∞ [N ]) ∧ such that ψ ≡ 0 (mod m K ) (where m K ⊆ O K is the maximal ideal). Let us write k for the residue ﬁeld O K /m K . Write ψ for ψ considered modulo m K . Then ψ is a section of L[E] ⊗N ⊗ k for some E. By enlarging K, we may assume that E consists solely of K-valued points of X K . Moreover, since ψ is nonzero, there exists some irreducible component of the special ﬁber of X[E]⊗k over which ψ is nonzero. If we choose p (in the above discussion) to be such that p ⊗ O K O K is this irreducible component, then it follows that the restriction of ψ ∈ (G ∞ [N ]) ∧ to Ω ⊗N L/K ⊗ K K will be nonzero. Now I claim that p satisﬁes the condition that κ L is not identically zero: Indeed, if κ L were identically zero, then it would follow that κ ⊗N would be identically L ⊗N zero. But the restriction of ψ to Ω ⊗ K K (which is assumed to be nonzero) is a nonzero L/K ⊗N K-multiple of κ L (ρ) (which would have to be zero). This contradiction proves the claim. Thus, we are in a position to apply (∗) L−pt . As discussed in the paragraph preceding this Proposition, it then follows that the restriction of ψ to Ω ⊗N L/K ⊗ K K is zero. Thus, we get a contradiction. This completes the proof of the Proposition. 64 Remark. In [Mzk2], where one only considers isomorphisms of π 1 ’s, as opposed to homo- morphisms as we do here, there is no need to place (as we did in (∗) L−pt ) the condition on p that κ L be not identically zero. Because in the present context it is necessary to include such a condition on p in (∗) L−pt , the author at ﬁrst did not see how it would be possible to prove the “preservation of relations” in the present context. However, in fact, in order to prove the preservation of relations (Proposition 12.1), it suﬃces to consider only p for which one knows already that κ L is not identically zero. This observation arose in discussions between the author and A. Tamagawa. Section 13: The Preservation of L-Points The purpose of this Section is to verify the assertion (∗) L−pt discussed in Section 12. The technique is similar to that employed in [Mzk2] (although we do not assume any knowledge of [Mzk2] in the following discussion). We continue with the notation of the preceding Section. In particular, we assume that we have been given a continuous open homomorphism over Γ K θ : Π U K → Π Y K Moreover, we assume that X K extends to a stable curve X → Spec(O K ). Let p be as in → X (that is, X is what we denoted by X[E] O (∗) L−pt . Thus, we have a blow-up X K → Spec(O K ). Choose a in Section 12), and p is a generic point of the special ﬁber of X smooth, aﬃne, geometrically connected open neighborhood W of p in the special ﬁber of Let T be the aﬃne scheme whose coordinate ring R T is such that Spf(R T ) (where we X. along W . Thus, T → Spec(O K ) equip R T with the p-adic topology) is the completion of X is (p-adically) formally smooth, and T ⊗ k = W . By abuse of notation, we shall write p for Let us write η T for the generic the prime of T that maps to the original p under T → X. point of T (regarded as a scheme). Let L be the quotient ﬁeld of the p-adic completion of (R T ) p . Thus, L is a p-adic ﬁeld whose residue ﬁeld is k(W ) (the function ﬁeld of W ). η T T L Note that we have natural morphisms ξ X : T → X; ξ X : η T → X K ; ξ X : Spec(L) → η T L X K ; ξ U : η T → U K ; ξ U : Spec(L) → U K . Let α η U T : Γ η T → Π U K be the morphism η T determined (up to conjugation by an element of Δ U ) by ξ U . Similarly, we have α L U : η T η T L L Γ L → Π U K . Composing α U , α U with θ gives α Y : Γ η T → Π Y K , α Y : Γ L → Π Y K . Now let Δ Y ⊆ Δ Y be a p-derivate (see Deﬁnition 0.2) of Δ Y . Thus, Δ Y is an open, characteristic subgroup of Δ Y . Note that α η Y T deﬁnes a section a η Y T : Γ η T → Π Y η of T Π Y η → Γ η T . Write Π Y η for the open subgroup Im(a η Y T ) · Δ Y ⊆ Π Y η . Thus, Π Y η gives T T T T rise to a ﬁnite étale covering Y η T → Y η T 65 Moreover, there exists some ﬁnite étale covering η T → η T such that Y Y η T × η T η T ∼ × K η T = Z K Y for some curve Z K → Spec(K ), where K is a ﬁnite extension of K contained in K(η T ). Let T be the normalization of T in η T . Then T → T is ﬁnite (since η T is of characteristic zero) and generically étale. By means of θ, we can pull-back the above covering to U η T : Thus, we obtain an open subgroup Π U η ⊆ Π U η , which corresponds to some ﬁnite étale covering U η T → U η T T T (which is geometrically connected over η T since Π U η contains the graph of α η U T in Π U η = T T Γ η T × Γ K Π U K ). This covering extends to a ﬁnite, possibly ramiﬁed covering X η T → X η T . Moreover, we have an isomorphism X X η T × η T η T ∼ × K η T = Z K X for some curve Z K → Spec(K ). By enlarging K , we may assume that this curve has a stable extension Z X → Spec(O K ). η T : η T → U K . Now it follows Next, let us go back to considering the morphism ξ U η T η T ) tautologically from the way we deﬁned α Y (i.e., the fact that it comes originally from ξ U η T η T η T that ξ U lifts naturally to a point ξ U : η T → U η T . Let ξ X : η T → X η T be the result of composing this morphism with U η T → X η T . Moreover, by restricting to η T , we get a point η η T X X ξ X T : η T → X η T . Projecting to Z K , we thus get a point ξ X : η T → Z K . Moreover, it Z T → Z X , where T ⊆ T is easy to see that this morphism extends to a morphism ξ Z X : T is an open subscheme obtained as the “D(f)” (i.e., the complement of V (f) ⊆ T ) for some f ∈ Γ(T , O T ) with the property that f is nonzero at every generic point of the special ﬁber of T → Spec(O K ). Indeed, this follows from the following two facts: (i) we η T X already have an extension to T K , since Z K → X K is ﬁnite, and ξ X : η T → X K extends to T K ; (ii) to extend from T K to some T , it suﬃces to apply the valuative criterion for properness (since Z X → Spec(O K ) is proper). Next, let us consider the morphism T ⊆ T → T . Clearly (after possibly enlarging K ) there exists a closed point t ∈ T such that if we let S be the completion of T at t, then the morphism S → S obtained by base-changing T → T by S → T has the following properties: (i) S → S is ﬁnite and ﬂat; (ii) S is a ﬁnite disjoint sum of connected components which are geometrically irreducible over O K ; (iii) each of these components admits a section over O K . Let S be any one of these connected components. Then we have the following: Lemma 13.1. The scheme S is normal. Proof. (Note that this is not entirely obvious since it is not clear that T is excellent.) First, I claim that the (ﬂat) morphism S → T is geometrically regular. Indeed, over the generic 66 ﬁber, this follows from the fact that S is regular and generically of characteristic zero; over a closed point of T K or T , either the ﬁber of the morphism S → T is (schematically) isomorphic to the given closed point, or it is empty. Thus, it remains to check what happens over the height one prime of T which is the special ﬁber of T → Spec(O K ). But over this prime, the geometric regularity of S → T follows from the excellence of T ⊗ k = W (which is ﬁnitely generated over a ﬁnite ﬁeld). This completes the proof of the claim. Thus, it follows that S → T is geometrically regular. Moreover, T is normal; hence, we obtain that S is normal, as desired. Thus, S → S satisﬁes the properties listed in Deﬁnition 6.3. Moreover, if we apply the base-change S → T to all the objects in the above discussion, it is easy to see that we are in the situation discussed in Proposition 7.4. Note, in particular, that the existence of the T morphism ξ Z → Z X shows that the condition (∗) S (cf. the discussion preceding X : T Deﬁnition 6.4) is satisﬁed by α η X S : Γ η S → Π X η . In fact, we also have the following: S Lemma 13.2. α η X S is F -geometric, hence also F I-geometric. T X Proof. Indeed, ﬁrst observe that restricting ξ Z . X gives rise to a morphism ξ 1 : S → Z Since S was constructed so as to admit a section over O K , let σ : Spec(O K ) → S be such a section. Then we may form the composite ξ σ : S → Z X of the structure morphism S → Spec(O K ) with ξ 1 ◦ σ. Now observe that ξ 1 and ξ σ both deﬁne sections def of Z S X = Z X × O K S → S (which coincide over Im(σ)). Let D 1 , D σ ⊆ Z S X be the Weil divisors which are the images of these two sections. Next, let us observe that by Lemma 13.3 below, these two Weil divisors are Q-Cartier. Hence it follows that there exists a positive integer N such that N · D 1 and N · D σ are, in fact, Cartier divisors, so we may def form the line bundle L = O Z X (N · D 1 − N · D σ ). Note that the line bundle L is trivial over S Im(σ) ⊆ S , hence over the closed point of S . Let J Z X → Spec(O K ) be the unique semi- X abelian scheme whose generic ﬁber is the Jacobian of Z K . Then it follows that L deﬁnes an S -valued point of J Z X which arises from an S -valued point of the formal completion of J Z X at the identity. But, sorting through the deﬁnitions (in particular, Deﬁnitions 6.2, 6.4) reveals that this implies that α η X S is F -geometric, hence also F I-geometric (cf. the paragraph following Deﬁntion 6.4). def Lemma 13.3. The scheme Z S X = Z X × O K S is normal, and, moreover, any Weil X divisor D ⊆ Z S arising as the graph of an O K -morphism ξ D : S → Z X is Q-Cartier. Proof. By Lemma 13.1 above, S is normal. Moreover, since Z S X is a generically smooth stable curve over S , one checks easily that the conditions “R 1 ” and “S 2 ” of Serre (see, e.g., [Mats], Chapter 7, §17.I, Theorem 39) are satisﬁed by Z S X , so Z S X is normal. Now recall that to say that a Weil divisor is “Q-Cartier” simply means that some nonzero multiple of that Weil divisor is Cartier. Next, let us observe that since the graph of ξ D is deﬁned by a 67 single equation in a neighborhood of any point of the smooth locus of Z X → Spec(O K ), it suﬃces to check that D is Q-Cartier in a neighborhood of the nodes of the special ﬁber of Z X → Spec(O K ). In fact, it even suﬃces to check that D is Q-Cartier over the spectrum of the completion of the local ring of Z S X at such a node. Thus, let us write A = Γ(S , O S ); def def B = O K [[x, y]]/(xy − π n ); def C = A[[x, y]]/(xy − π n ) where π ∈ O K is a uniformizer, and n is a positive integer. Here, we think of B (respec- tively, C) as the result of completing Z X (respectively, Z S X ) at a node (respectively, at the inverse image of this node via the projection Z S X → Z X ). Recall that S is aﬃne, so S = Spec(A). Now ξ D is given by some morphism B → A. Write s x , s y ∈ A for the images of x and y, respectively, under this morphism. Note that s x · s y = π n ∈ A. Moreover, the restriction D C of the subscheme D to Spec(C) is deﬁned by the equations x − s x , y − s y , i.e., D C = V (x − s x , y − s y ). Now let us consider the regular function x − s x on Spec(C). I claim that the ideals (x − s x ) and (x − s x , y − s y ) in C coincide in C[ π 1 ]. Indeed, this follows from the following equation: y − s y = −y · s y 1 (x − s x ) ∈ C[ ] n π π Thus, we conclude that on Spec(C), the Weil divisor D C is linearly equivalent to a Weil divisor E on Spec(C) which is supported on the special ﬁber F C of Spec(C) → Spec(O K ). On the other hand, the fact that S → Spec(O K ) is geometrically irreducible implies that (F C ) red has precisely two irreducible components, namely, V (x) red and V (y) red . Thus, it follows that any divisor supported on F C is Q-Cartier, as desired. This completes the proof of the Lemma. Thus, (by Lemma 13.2) we may apply Proposition 7.4 to conclude that Y η S admits a line bundle of degree prime to p. Moreover, by Proposition 8.1 and Lemma 8.2, it thus follows that: def Y L = Y η T × η T Spec(L) has a rational point over some tamely ramiﬁed extension L of L. It is this key result that will allow us to conclude the proof of (∗) L−pt . Let us review what we have done so far, from the point of view of objects over Y L . L First, we have a section a L Y : Γ L → Π Y L of Π Y L → Γ L (deﬁned by α Y ). Moreover, given def any p-derivate Δ Y ⊆ Δ Y , we obtain a ﬁnite étale covering Y L → Y L deﬁned by Π Y = L L Im(a L Y ) · Δ Y ⊆ Π Y L . Under these circumstances, we just showed that Y L necessarily has L 68 a rational point over some tamely ramiﬁed extension of L. Moreover, tracing through the deﬁnitions, it is easy to see that the assumption placed on p in (∗) L−pt that κ L be not identically zero means precisely (in the language introduced at the beginning of Section 9) that a L Y is nondegenerate. It thus follows that we can apply Corollary 10.5 to conclude that: L a L Y : Γ L → Π Y L arises from some geometric point (ξ ) ∈ Y L (L). In other words, the proof of (∗) L−pt has been completed. Thus, by Proposition 12.1, we conclude the following: Corollary 13.4. Let θ : Π U K → Π Y K be a continuous open homomorphism over Γ K . Then θ “preserves relations,” i.e., (in the notation of Section 12), we have Ψ N (R N ) = 0, for all positive integers N . Note that in [Mzk2] (where we essentially dealt with the case where θ is an isomor- phism), the preservation of relations is already enough to conclude the proof of the main theorem. In the present context, however, because of the fact that H U is of inﬁnite rank, it is necessary to go through one more intermediate technical step before we can complete the proof of the (ﬁrst part of the) ﬁrst main theorem. This step essentially amounts to showing that any θ as in Corollary 13.4 necessarily factors through Π X K . The proof of this next step is the main topic of the following Section. Note that if this fact (i.e., that θ factors through Π X K ) could be proven more directly, then this paper could be simpliﬁed con- siderably. (For instance, Section 11, as well as the rather technical notions of “irreducibly splittable” and “F I-geometric” would be unnecessary.) Unfortunately, however, the proof of this fact in Section 14 relies heavily on the “preservation of relations” (Corollary 13.4). Section 14: The Annihilation of Inertia In this Section, we prove that any continuous surjective homomorphism θ : Π U K → Π Y K over Γ K necessarily (acts as though it) factors through Π X K . In the process of doing this, we complete the proof of the (ﬁrst part of) the ﬁrst main theorem of this paper, in the case where the base ﬁeld is a local ﬁeld. Throughout most of this Section (except for the very end – i.e., from the statement of Theorem 14.1 on), we continue to use the notation of the preceding three Sections. Let us assume, moreover, that Y K is not hyperelliptic. Thus, it follows (e.g., from Lemma 10.4 (3)) that any connected ﬁnite étale covering of Y K is also non-hyperelliptic. Let us ﬁrst consider the morphism → F ∧ H 0 (Y, ω Y /O K ) ⊗ O K O ∞ K 69 constructed in Section 12. We would like to show ﬁrst of all that F F ⊆ F ∧ , where F F def ⊗ O K O (∗) F X This morphism factors through F X ∞ X = K 0 H (X, ω X/O K ) (cf. the discussion preceding Proposition 11.3). To do this, we argue as follows: Let x ∈ X K (K). Let ∧ x : F ∞ → O K be the morphism induced by restriction: i.e., elements of F ∞ are sections of L[∞], hence diﬀerentials on X[∞]; thus, by taking the residue of such a diﬀerential at x, we get a residue map F ∞ → O K ; then taking the p-adic completion of this residue map gives us x . We would like to show in the following that the restriction ζ x : H 0 (Y, ω Y /O K ) → O K of x to H 0 (Y, ω Y /O K ) is zero. If we show this, then this will be also hold for all x ∈ F X K (K ) (where K is a ﬁnite extension of K), hence (∗) F X will follow immediately (from F consists precisely of all those diﬀerentials whose residues at every point the fact that F X are zero). Thus, let us assume that ζ x = 0. Let I x ⊆ Δ U denote the inertia group (well-deﬁned up to conjugation) corresponding to x. Then observe that the restriction I x ⊆ Δ U → Δ Y of θ to Δ U is nontrivial. Indeed, if this restriction were zero, then it is clear from the way that ζ x was constructed (cf. the “comparison theorem” of Proposition 11.3) that ζ x would be zero. Thus, θ(I x ) = {1}. In particular, it follows (by possibly enlarging K) that there exists an open normal subgroup Π Y ⊆ Π Y K (corresponding to a covering Y K → Y K ) such K that: (i) Π Y surjects onto Γ K ; and (ii) if we let U K → U K be the result of pulling back K Y K → Y K via θ, then U K → U K is a connected Galois covering which is ramiﬁed over x. def Let G = Π Y K /Π Y . Thus, G is a ﬁnite group, and G = Gal(U K /U K ) = Gal(Y K /Y K ). K Note that U K → U K extends to a ramiﬁed covering X K → X K . Let x ∈ X K (K) (where we enlarge K if necessary) be a point lying above x. Now let us denote with primes the objects corresponding to U K , X K , and Y K that are analogous to the objects already constructed for U K , X K , and Y K . Thus, we have ∧ x : (F ∞ ) → O K and ζ x : H 0 (Y , ω Y /O K ) → O K 70 Note that the fact that ζ x = 0 implies (since ζ x | H 0 (Y,ω Y /O K ) = ζ x ) that ζ x = 0. Let σ ∈ G be such that σ(x ) = x , σ = 1. Then the fact that σ ﬁxes x implies that σ ﬁxes x , ∧ ) is G-equivariant) which, in turn, implies (since the inclusion H 0 (Y , ω Y /O K ) → (F ∞ that σ ﬁxes ζ x . On the other hand, by Corollary 13.4 (“preservation of relations”) and the fact def that Y K is not hyperelliptic, it follows that the point of the projective space P K = 0 P(H (Y , ω Y /O K )) deﬁned by ζ x lies on the canonically embedded curve Y K ⊆ P K . Thus, the fact that ζ is ﬁxed by σ implies that Y admits a K-valued point which is ﬁxed x K by σ. Since Y K → Y K = Y K /G is étale, however, this is absurd. This contradiction thus F completes the proof that ζ x = 0, and hence also the proof of (∗) F X . Let us review what we have done so far. Given a surjective continuous homomorphism θ : Π U K → Π Y K over Γ K , we have seen that θ induces an injection H 0 (Y K , ω Y K /K ) → H 0 (X K , ω X K /K ) that preserves relations. Thus, it follows (by using the canonical embedding of Y K ) that we get a dominant morphism θ U : U K → Y K which extends to a morphism θ X : X K → Y K by properness. Moreover, given any ﬁnite Galois étale covering Y K → Y K whose geometric part has p-power order, we can pull-back this covering via θ to obtain U K → U K (ﬁnite étale), X K → X K (ﬁnite and possibly ramiﬁed), together with θ : Π U → Π Y . If K K we then repeat the argument just applied to θ for θ , we see that we get a morphism θ X If we then continue this procedure for arbitrary : X K → Y K which lies over θ X . ﬁnite Galois étale coverings (whose geometric parts have p-power order), the well-known correspondence between fundamental groups and categories of étale coverings thus shows that the morphism induced by θ U on fundamental groups coincides with θ up to composition with an inner automorphism induced by an element of Π Y K . On the other hand, since both θ and the morphism induced by θ U on fundamental groups have the property that they lie over Γ K , it follows that the element of Π Y K in question must map to the the center of Γ K . Since, however, Γ K is center-free (see Lemma 15.6 below – one checks easily that there are no “vicious circles” in the reasoning), it follows that the element in question must therefore be an element of Δ Y ⊆ Π Y K . That is to say, we have essentially proven the following result: Theorem 14.1. Let K be a ﬁnite extension of Q p . Let Y K be a hyperbolic curve (not necessarily proper) over K. Let U K be the spectrum of a one-dimensional function ﬁeld over K. Let Hom dom K (U K , Y K ) be the set of dominant K-morphisms from U K to Y K . Let (Π , Π Hom open U K Y K ) be the set of open, continuous group homomorphisms Π U K → Π Y K Γ K over Γ K , considered up to composition with an inner automorphism arising from Δ Y . Then the natural map 71 open Hom dom K (U K , Y K ) → Hom Γ K (Π U K , Π Y K ) is bijective. Proof. We begin by proving that this map is injective. First, observe that by replacing U K and Y K by coverings deﬁned by “p-derivates” (as in Deﬁnition 0.2) of Δ U and Δ Y (where we use p-derivates since they are natural), we may assume that both U K and Y K admit hyperbolic compactiﬁcations U K and Y K (i.e., U K and Y K are proper hyperbolic curves over K). Since dominant K-morphisms U K → Y K are the same as dominant K-morphisms U K → Y K , it suﬃces to show that such a morphism U K → Y K is determined by its induced morphism Δ ab → Δ ab (on abelianizations of geometric fundamental groups). But U Y this follows from the fact that this morphism Δ ab → Δ ab clearly determines the morphism U Y between all p-power torsion points of the Jacobians of U K and Y K , hence it determines the induced morphism between the Jacobians of U K and Y K . Moreover, since U K and Y K are both proper hyperbolic (i.e., of genus ≥ 2), they both embed in their Jacobians, so we conclude that the original morphism U K → Y K is uniquely determined, as desired. Next, we consider surjectivity. Let us ﬁrst consider the case where Y K is proper. By replacing U K by a ﬁnite étale covering of U K (and then descending at the end, which is possible since (by the preceding paragraph) the natural map in the Theorem is injective), we can assume that the proper model X K of U K is hyperbolic. Note, moreover, that if θ : Π U K → Π Y K is open, then its image is of ﬁnite index, so by replacing Y K by a ﬁnite étale covering of Y K , it is clear that we may assume that θ is surjective. Finally, by Lemma 10.4 (4), by replacing Y K by a ﬁnite étale covering of Y K , it is clear that we may assume that Y K is non-hyperelliptic. Now we are in the circumstances considered above, and so we see that θ arises from a geometric morphism U K → Y K , as desired. This completes the proof of the Theorem when Y K is proper. Now let us consider the case when Y K is not proper. First note that by replacing Y K by a ﬁnite étale covering of Y K , we may assume that the compactiﬁcation Z K of Y K is hyperbolic. Now the point is to compose the given θ : Π U K → Π Y K with the morphism Π Y K → Π Z K arising from the compactiﬁcation inclusion Y K ⊆ Z K . Since we know the Theorem to be true for morphisms between U K and Z K , we thus obtain that Π U K → Π Z K arises from some dominant U K → Z K (which necessarily factors – since U K is the spectrum of a ﬁeld – through Y K , thus yielding a U K → Y K ). Thus it remains only to see that the morphism induced on π 1 ’s by this U K → Y K is the same as the given θ. But this is done (as usual) by considering a ﬁnite étale covering Y K → Y K , and applying the argument just described to U K and Y K to obtain a U K → Y K which lies over the U K → Y K constructed previously. As usual, this is enough to show (cf. the argument directly preceding the statement of Theorem 14.1) that the morphism induced on fundamental groups by the U K → Y K that we constructed is the same as the given θ. This completes the proof of the Theorem. In fact, we have the following: 72 Corollary 14.2. Let K be a ﬁnite extension of Q p . Let X K (respectively, Y K ) be either a hyperbolic curve (not necessarily proper!) over K or the spectrum of a one-dimensional function ﬁeld over K. Let Hom dom K (X K , Y K ) be the set of dominant K-morphisms from X K to Y K . Let Hom open (Π , Π X K Y K ) be the set of open, continuous group homomorphisms Γ K Π X K → Π Y K over Γ K , considered up to composition with an inner automorphism arising from Δ Y . Then the natural map open Hom dom K (X K , Y K ) → Hom Γ K (Π X K , Π Y K ) is bijective. Proof. By an argument analogous to that of the ﬁnal paragraph of the proof of Theorem 14.1, it follows that it suﬃces to consider the case where Y K is a hyperbolic curve. If U K is the generic point (considered as a scheme) of X K , then given any open θ X : Π X K → Π Y K , composing θ X with the natural surjection Π U K → Π X K induced by U K → X K gives us an open θ U : Π U K → Π Y K . Applying Theorem 14.1 to θ U thus gives us a dominant morphism X K → Y K (where Y K is the compactiﬁcation of the curve Y K ). To see that this morphism factors through Y K , it suﬃces to apply this construction to Y K → Y K , where Y K is a ﬁnite, geometrically connected (over K), Galois étale covering (of p-power order) of Y K which is ramiﬁed over all the points of Y K − Y K . In fact, we may even choose Y K so that the ramiﬁcation indices over all the points of Y K − Y K are larger than the degree of X K over Y K . Then the fact that we get some X K → Y K (where X K is étale over X K ) lying over the morphism X K → Y K obtained previously shows that this morphism X K → Y K factors through Y K , as desired. This completes the proof of the Corollary. Remark. Corollary 14.2 is thus a special case of Theorem A, the ﬁrst main theorem of this paper. In fact, Corollary 14.2 holds even in the case where K is only ﬁnitely generated over Q p . However, unlike the situation in [Mzk2], deriving this ﬁnitely generated case from the local ﬁeld case is not so trivial, again (cf. the discussion at the end of Section 13) because of the fact that Δ U is so large. Thus, we save the derivation of the ﬁnitely generated case for the following Section. Section 15: Base Fields Finitely Generated over the p-adics In this Section, we let L be a ﬁnitely generated extension of Q p . We would like to prove versions of Theorem 14.1 and Corollary 14.2 in the case where the base ﬁeld is L (as opposed to a ﬁnite extension of Q p ). Thus, by induction on the transcendence degree of L over Q p , we may assume that Theorem 14.1, for instance, is known for all base ﬁelds whose transcendence degree over Q p is < that of L. Let X L and Y L be proper hyperbolic curves over L. Let U L be the generic point of X L (regarded as a scheme). Moreover, let us assume that we have been given a continuous surjective homomorphism over Γ L 73 θ : Π U L → Π Y L Once it is shown that θ factors though Π X L , it is relatively easy to derive that θ arises geometrically from Theorem 14.1. Thus, the ﬁrst order of business is to show that θ factors through Π X L . To achieve this, we assume that this is not the case. Then (cf. the argument employed in Section 14) by replacing Y L and U L by ﬁnite étale coverings Y L → Y L and U L → U L (the latter obtained by pulling back the former via θ), we may assume that we are in the following situation: There is a ﬁnite cyclic group G (with generator σ) acting faithfully on U L (hence also X L ) and Y L – via L-linear automorphisms – in such a way that σ ﬁxes a point x 0 ∈ X L (L), but acts without ﬁxed points on Y L . Moreover, we assume that σ ◦ θ ◦ σ −1 coincides with θ up to composition with an inner automorphism deﬁned by an element of Δ Y . If we can show that these assumptions lead to a contradiction, it will follow immediately that θ factors through Π X L . Let (as usual) H U (respectively, H Y ) be the abelianization of Δ U (respectively, Δ Y ) regarded as a Γ K -module. Let H U P ⊆ H U be the closure of the image of all the inertia groups (cf. the beginning of Section 11). Note that θ induces a Γ K -morphism H U → H Y . Since H Y is a ﬁnitely generated Z p -module, it follows that (by possibly replacing L by a ﬁnite extension of L) there exist points x 1 , . . . , x r ∈ X L (L) such that the images of the corresponding inertia groups θ(I i ) (for i = 1, . . . , r) in H Y is equal to the image of θ(H U P ) in H Y . Now let K ⊆ L be a subﬁeld such that L is a one-dimensional function ﬁeld over K (hence, in particular, we assume that K is algebraically closed in L). Note that such a K always exists (as long as L is not a ﬁnite extension of Q p ). Thus, there exists a smooth aﬃne model M → Spec(K) of L such that X L and Y L extend to smooth curves X M → M and Y M → M over M. Moreover, we may also assume (by shrinking M) that x 1 , . . . , x r extend to sections s 1 , . . . , s r : X M → M whose images are disjoint from one another, and that σ acts on X M and Y M . The point x 0 ∈ X L (L) then extends (by the valuative criterion for properness) to a section s 0 : M → X M . Finally, let us observe that (by further shrinking M), we may assume that σ acts without ﬁxed points on Y M . The ﬁrst thing that we would like to get our hands on is a morphism like θ, except for objects over K, so that we can apply Theorem 14.1 over K (which we know to be true via the induction hypothesis). To construct such a morphism, we argue as follows. Let m ∈ M(K) be a point (which, after possibly enlarging K, always exists). Let D be the spectrum of the completion of O M,m . (Here, one should think of the “D” as standing for “disk.”) Thus, we may choose an isomorphism D ∼ = Spec(K[[t]]) (where t is an indeterminate). Let D ∞ → D 1 be the ramiﬁed covering of inﬁnite degree obtained by adjoining all t N (for N a positive integer) to O D . Let p ∈ X M be the prime which is the ﬁber X m of X M → M over M. Let U D be the spectrum of the completion of O X M , p . Thus, U D is the spectrum of a complete discrete valuation ring with residue ﬁeld equal to K(U m ), the function ﬁeld of X m . Let η D def (respectively, η D ∞ ) be the generic point of D (respectively, D ∞ ). Let U η D = U D × D η D . Thus, U η D is the spectrum of a complete, discretely valued ﬁeld. In particular, it follows 74 from the well-known theory of such ﬁelds (see, e.g., [Ser2]) that if Γ U ηD is the absolute Galois group of this ﬁeld, then we have an exact sequence 1 → Z(1) → Γ U ηD → Γ K(U m ) → 1 Here the covering corresponding to the Z(1) is given by adjoining all t N (for N a positive integer) – cf. the covering D ∞ → D. 1 Now let us denote by Π U η the result of replacing the geometric portion (i.e., relative D to the morphism U η D → η D ) of the fundamental group of U η D by its maximal pro-p quotient. Then we get an exact sequence 1 → Δ U m → Π U η D → Γ η D → 1 If we pull this exact sequence back via Γ η D ∞ = Γ K → Γ η D , we thus get an exact sequence 1 → Δ U m → Π U η D ∞ → Γ η D ∞ = Γ K → 1 Moreover, it is easy to see that this last exact sequence can be identiﬁed with 1 → Δ U m → Π U m → Γ K → 1 On the other hand, if we pull back Π Y L → Γ L via Γ K = Γ η D ∞ → Γ L , we get an exact sequence 1 → Δ Y m → Π Y m → Γ K → 1 Thus, if we pull-back θ via Γ K = Γ η D ∞ → Γ L to obtain a morphism Π U η D ∞ → Π Y η D ∞ (where the subscripted η D ∞ denotes “⊗ L η D ∞ ”) and compose with “π 1 ” of the natural morphism U η D ∞ → U η D ∞ , we get a continuous homomorphism θ m : Π U m → Π Y m over Γ K (where K we regard here as the residue ﬁeld of M at m). Lemma 15.1. The morphism θ m is surjective. Proof. It suﬃces to show that the restriction of θ m to Δ U m surjects onto Δ Y m = Δ Y . In fact, by the basic theory of p-groups, it suﬃces to show that θ m induces a surjection of Δ U m onto H Y m = H Y . To see this, it suﬃces to consider (after replacing M by a ﬁnite 75 étale covering of M, and enlarging K if necessary) a ﬁnite abelian covering Y L → Y L (where Y L → L is geometrically connected) of degree a power of p which is > 1. Let us pull-back this covering to some covering U L → U L via θ. Then we must show that the pull-back U η D ∞ → U η D ∞ of this covering to U η D ∞ is not the trivial covering. Thus, suppose that it is the trivial covering. Then the covering U m → U m that it induces (cf. the exact sequences above) of U m is again trivial. Since the images of s 1 , . . . , s r in X m are disjoint this implies ﬁrst of all that U L → U L is unramiﬁed at x 1 , . . . , x r . But because of the way in which x 1 , . . . , x r were chosen, this implies that the covering Y L → Y L was obtained from a quotient of H Y /Im(θ(H U P )). Thus, it follows that U L → U L extends to a ﬁnite étale covering X L → X L . Moreover, since X M is smooth over M at m, it follows that X L → X L is split if and only if the induced X m → X m is split. But this X m → X m extends the covering U m → U m which we already saw to be trivial. Thus, we obtain that X L → X L , hence U L → U L is trivial. Since θ is surjective, however, this implies that Y L → Y L is trivial. This contradiction completes the proof of the Lemma. Now by the induction hypothesis on the transcendence degree of L, it follows that θ m arises from some geometric morphism U m → Y m which is compatible with the action of σ on both sides (since σ is compatible with θ). By the valuative criterion for properness, this morphism extends to a morphism X m → Y m . Moreover, σ ﬁxes s 0 (m) ∈ X m (K), so s 0 (m) ∈ X m (K) is mapped to a ﬁxed point of Y m (K), which is absurd, since σ acts on Y m without ﬁxed points. This contradiction completes the proof of the following result: Lemma 15.2. Let L be a ﬁnitely generated extension of Q p . Let X L and Y L be proper hyperbolic curves over L. Let U L be the generic point of X L . Then any continuous surjective homomorphism θ : Π U L → Π Y L over Γ L necessarily factors through Π X L . Now we can conclude that θ arises geometrically, as follows. Consider the M-scheme def H M = Hom M (X M , Y M ) → M. Since Y M is hyperbolic, it is well-known that H M is ﬁnite and unramiﬁed over M. (Indeed, that H M → M is unramiﬁed (respectively, quasi- ﬁnite; proper) follows since the pull-back of the tangent bundle of Y m to X m (for any m ∈ M) has no global sections (respectively, follows from the Hurwitz formula, which allows one to bound the degree of a morphism X m → Y m ; follows via the same argument as that used in Lemma 8.3 of [Mzk1]).) By shrinking M, we may assume that H M is ﬁnite étale over M. Then the fact that θ m arises geometrically (from some X m → Y m ) shows that over some ﬁnite extension L of L, we have a morphism X L → Y L that specializes to X m → Y m . Moreover, if M is the normalization of M in L , then since the pro-p geometric fundamental groups of X M and Y M form local systems over M , it follows (by checking what happens over the point m) that the morphism on Δ’s induced by X L → Y L is the same (up to composition with an inner automorphism) as that induced by θ. In particular, the morphism H X → H Y induced by X L → Y L is the same as that induced by θ. On the other hand, a morphism from X L to Y L (over any ﬁeld) is determined by the morphism it induces from H X to H Y (cf. the proof of the injectivity part of Theorem 14.1). Thus, since the H X → H Y in question is Γ L - (not just Γ L -) equivariant, we obtain 76 that X L → Y L descends to a X L → Y L . By repeating this construction (as usual) for all ﬁnite étale coverings of Y L , we also obtain that the morphism induced on fundamental groups by this X L → Y L is the original θ. That is to say, in summary, we have proven that any θ : Π U L → Π Y L as in Lemma 15.2 necessarily arises geometrically from some morphism U L → Y L . Thus, by arguments formally analogous to those of the proofs of Theorem 14.1 and Corollary 14.2, we obtain the following analogue of Corollary 14.2: Corollary 15.3. Let K be a ﬁnitely generated ﬁeld extension of Q p . Let X K (re- spectively, Y K ) be either a hyperbolic curve over K or the spectrum of a one-dimensional function ﬁeld over K. Let Hom dom K (X K , Y K ) be the set of dominant K-morphisms from X K to Y K . Let Hom open (Π , Π X K Y K ) be the set of open, continuous group homomorphisms Γ K Π X K → Π Y K over Γ K , considered up to composition with an inner automorphism arising from Δ Y . Then the natural map open Hom dom K (X K , Y K ) → Hom Γ K (Π X K , Π Y K ) is bijective. It turns out that it is most natural to generalize Corollary 15.3 to the case where the long and unwieldy expression “hyperbolic curve/spectrum of a one-dimensional function ﬁeld deﬁned over a ﬁnitely generated ﬁeld extension of Q p ” is replaced by the much shorter expression “pro-hyperbolic curve over a sub-p-adic ﬁeld.” Thus, we make the following deﬁnition: Let K be a ﬁeld. Let X K be a K-scheme. Deﬁnition 15.4. (i) We shall call a ﬁeld K a sub-p-adic ﬁeld if there exists a prime num- ber p, together with a ﬁnitely generated ﬁeld extension L of Q p such that K is isomorphic to a subﬁeld of L. (ii) We shall call X K a hyperbolic pro-curve (over K) if X K can be written as the projec- tive limit of a projective system of hyperbolic curves over K such that all the transition morphisms in the projective system are birational. Thus, the following are all examples of sub-p-adic ﬁelds: (1) ﬁnitely generated (in particular, ﬁnite) extensions of Q p (2) number ﬁelds (i.e., ﬁnite extensions of Q) (3) the subﬁeld of Q which is the composite of all number ﬁelds of degree ≤ n over Q (for some ﬁxed integer n). 77 Another way to think of a pro-hyperbolic curve X K is as the result of removing some set (possibly inﬁnite, possibly empty) of closed points from some hyperbolic curve. In partic- ular, the notion of a “hyperbolic pro-curve” generalizes both the notion of a hyperbolic curve and the case of the spectrum of a function ﬁeld of dimension one. Then we have the following result: Corollary 15.5. Let K be sub-p-adic. Let X K (respectively, Y K ) be a hyperbolic pro-curve over K. Let Hom dom K (X K , Y K ) be the set of dominant K-morphisms from X K open to Y K . Let Hom Γ K (Π X K , Π Y K ) be the set of open, continuous group homomorphisms Π X K → Π Y K over Γ K , considered up to composition with an inner automorphism arising from Δ Y . Then the natural map open Hom dom K (X K , Y K ) → Hom Γ K (Π X K , Π Y K ) is bijective. Proof. First, let us observe that if K is a ﬁnitely generated ﬁeld extension of Q p , then Corollary 15.5 follows immediately from Corollary 15.3 by arguments formally analogous to those of the proofs of Theorem 14.1 and Corollary 14.2. Thus, the only (slightly) “new” phenomenon here is the fact that we allow K to be a subﬁeld of a ﬁnitely generated extension of Q p . The argument for such subﬁelds is as follows: Let K be a subﬁeld of a ﬁnitely generated extension ﬁeld L of Q p . Suppose that we have been given X K and Y K as in the statement of Corollary 15.5, as well as an open θ : Π X K → Π Y K over Γ K . Let us denote by θ geom the morphism Δ X → Δ Y induced by θ on geometric fundamental groups. By base-changing to L and applying the Corollary 15.5 over L, we obtain that there exists a morphism φ L : X L → Y L whose induced morphism on geometric fundamental groups coincides with that deﬁned by θ. On the other hand, since morphisms between hyperbolic curves (hence also hyperbolic pro-curves), clearly have no moduli (cf. the fact that the scheme H M of the discussion preceding Corollary 15.3 was ﬁnite and unramiﬁed over M), it follows that φ L descends to a ﬁnite Galois extension K of K. Thus, we have a morphism φ K : X K → Y K . It remains to descend φ K to K. But this follows from the fact that if σ ∈ Gal(K /K), then conjugating φ K by σ gives a morphism φ σK : X K → Y K whose induced morphism on geometric fundamental groups is the result of conjugating θ geom by σ. On the other hand, θ geom arises from θ which lies over Γ K , so θ geom is ﬁxed (up to composition with an inner automorphism deﬁned by an element of Δ Y ) by conjugation by σ. Thus, φ K and φ σK induce the same morphism on geometric fundamental groups, hence coincide (cf. the argument of the discussion preceding Corollary 15.3). This shows that φ K descends to a morphism φ K : X K → Y K . Repeating this construction (as usual – cf. the argument preceding the statement of Theorem 14.1) for coverings of X K and Y K shows that the morphism induced by φ K on fundamental groups coincides with θ (up to composition with an inner automorphism deﬁned by an element of Δ Y ). Remark. Note that in Corollaries 15.3 and 15.5, in fact, we implicitly used the fact that 78 for K as in those Corollaries, Γ K is center-free. This may be proven as follows. First, if K is a ﬁnite extension of Q p , then we have the following: Lemma 15.6. If K is a ﬁnite extension of Q p , then the group Γ K is center-free. Proof. (The argument given here is “well-known,” but was related to the author by A. Tamagawa.) Since one knows explicitly (see, e.g., [Ser2], Chapter IV, §2) the structure of Gal(K tm /K), it is clear that the quotient Gal(K tm /K) of Γ K is center-free. Thus, the center of Γ K must lie in Γ K tm ⊆ Γ K . On the other hand, Γ K tm is a pro-p group. Moreover, it follows immediately from the facts that (1) H 2 (K, F p (1)) = F p ; (2) if K is a ﬁnite unramiﬁed extension of K of degree p (which always exists), then the natural morphism H 2 (K, F p (1)) → H 2 (K , F p (1)) is zero; that H 2 (Γ K tm , F p ) = 0. But by [Shatz], Chapter III, §3, Proposition 2.3, this is enough to imply that Γ K tm is free (as a pro-p group), which implies that its center is trivial (see, e.g., [Tama], §1, Propositions 1.1, 1.11). Next, we consider the case of a ﬁnitely generated extension of Q p : Lemma 15.7. Suppose that K 0 is a ﬁeld of characteristic zero with the property that every open subgroup of Γ K 0 is center-free. Then any ﬁnitely generated ﬁeld extension K of K 0 also has this property. In particular, if K is a ﬁnitely generated ﬁeld extension of Q p , then Γ K is center-free. Proof. The last statement follows from the ﬁrst plus Lemma 15.6. Thus, let us prove the ﬁrst statement. Note that an extension of a center-free group by a center-free group is center-free. Thus, it suﬃces to prove that if L is a function ﬁeld (of arbitrary ﬁnite dimension) over K 0 , then Γ L is center-free. Next, note that the projective limit of a projective system of center-free groups in which all the transition morphisms are surjective is center-free. Moreover, as is well-known, Spec(L) is a projective limit of “hyperbolic Artin neighborhoods” over K 0 (i.e., successive ﬁbrations of hyperbolic curves – see [SGA4], XI 3.3). Thus, it suﬃces to prove that the fundamental group of such an Artin neighborhood is center-free. But this then reduces to showing that the fundamental group of a hyperbolic curve is center-free, which is well-known (see, e.g., [Tama], §1, Proposition 1.11). This is already enough for Corollary 15.3. Now we can conclude the result for arbitrary K as in Corollary 15.5 by means of the following Lemma (due to A. Tamagawa): 79 Lemma 15.8. Let K be sub-p-adic. Then Γ K is center-free. Proof. Let K be a ﬁnitely generated ﬁeld extension of Q p that contains K. Suppose that σ ∈ Γ K lies in the center of Γ K , but is not equal to the identity. Then there exists a ﬁnite Galois extension L of K such that σ maps to an element of Gal(L/K) other than the identity. Write L = K(α), for some α ∈ L. Let E L be an elliptic curve over L with j-invariant α. Let X L be the complement of “0” in E L . Thus, X L is a hyperbolic curve over L. Let Y L be the result of base-changing X L by σ : L → L. Thus, it follows that X L and Y L are isomorphic as schemes over K. Moreover, conjugating by σ deﬁnes an isomorphism Π X L ∼ = Π Y L which lies over Γ L because σ is in the center of Γ K . Base- changing this isomorphism to L (the composite of K and L over K), we get a Γ L - isomorphism Π X L ∼ = Π Y L . By Corollary 15.3 (in fact, really, this follows already from the results of [Mzk2]), we obtain that this isomorphism arises from an L -isomorphism of X L with Y L . But this implies that the j-invariants of the compactifying elliptic curves of X L and Y L are the same, i.e., that α = α σ ∈ L ⊆ L , which is absurd. This contradiction completes the proof of the Lemma. Section 16: Maps From Higher-Dimensional Function Fields to Curves Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let U K be the spectrum of a function ﬁeld over K. (Note that here, we do not assume that the dimension of the function ﬁeld is one.) Let X K be a smooth projective model of U K (which exists by [Hiro]). Thus, U K is the generic point of X K . Let n be the dimension of X K . By abuse of terminology, we shall also say that n is the “dimension of U K over K.” Since the one-dimensional case has been dealt with previously, we assume here that n ≥ 2. The purpose of this Section is to prove a result like Corollary 15.5, except for morphisms between U K and a hyperbolic pro-curve. Let L be a very ample line bundle on X K . Let def V = Γ(X K , L) Since the dimension of X K is ≥ 2, and L is very ample, it follows that dim K (V ) ≥ 3. Let W ⊆ V be a one-dimensional (over K) subspace, generated by a section of L whose zero locus forms a smooth, connected subvariety of X K . (Note that by Bertini’s theorem (see, e.g., Theorem 6.3, pp. 66, of [Jou]), it is well-known that such a W exists.) Let W ⊆ V be a two-dimensional subspace containing W . Choose a basis {e 1 , e 2 } for W such that e 1 ∈ W . Let e 3 ∈ V be an element whose image in V /W is nonzero. Let us also assume that the common zero locus in X K of the three sections e 1 , e 2 , e 3 is a subscheme of X K of codimension ≥ 3. (This can always be achieved by choosing e 1 , e 2 , e 3 suﬃciently def def generically.) Let K = K(t) (where t is an indeterminate). Let s 1 = t; s 2 = t −1 . Then we def can construct a new two-dimensional subspace W ⊆ V K = V ⊗ K K , over K , as follows: 80 def def We let W be the subspace generated by the vectors e 1 = e 1 + s 1 · e 3 , e 2 = e 2 + s 2 · e 3 . Thus, to summarize, we have the following situation: W ⊆ ⊆ W K · e 1 + K · e 2 + K · e 3 K · e 1 ⊆ V K · e 1 + K · e 2 ⊆ K · e 1 + K · e 2 + K · e 3 e 2 ⊆ V K = V ⊗ K K ; W = K · e 1 + K · e 1 = e 1 + s 1 · e 3 ; ⊆ def e 2 = e 2 + s 2 · e 3 ; K = K(t) s 1 = t; s 2 = t −1 Lemma 16.1. Let Ω be an algebraically closed ﬁeld containing K . Then there do not def exist any one-dimensional subspaces W of W Ω = W ⊗ K Ω such that W is deﬁned (as a subspace of V ⊗ K Ω) over a ﬁnite extension of K. Proof. Indeed, if such a W existed, then there would exist elements a, b, c ∈ Ω (with c = 0, and at least one of a, b nonzero), together with α 1 , α 2 , α 3 ∈ K such that a · e 1 + b · e 2 = c · (α 1 · e 1 + α 2 · e 2 + α 3 · e 3 ) By dividing a and b by c, we may assume that c = 1. Then, equating the coeﬃcients of e 1 , e 2 , e 3 , we obtain that a = α 1 ; b = α 2 ; s 1 · a + s 2 · b = α 3 . But this implies that t, t −1 , and 1 (as elements of K ) are linearly dependent over K, which is absurd. This completes the proof of the Lemma. Before proceeding, let us pause to interpret what this Lemma means. Note that the 3-dimensional K-subspace of V generated by e 1 , e 2 , e 3 deﬁnes a rational morphism ψ from def X K to Q K = P(K · e 1 + K · e 2 + K · e 3 ). Thus, ψ is deﬁned outside of some closed subscheme of X K of codimension ≥ 2. Moreover, note that the inverse image via ψ of any closed point of Q K is a subscheme of X K of codimension ≥ 2. (Indeed, this follows from the fact that the common zero locus of e 1 , e 2 , e 3 has codimension ≥ 3 in X K .) Now let Ω be a ﬁnite extension of K , and let us consider a one-dimensional (over Ω) subspace W def of W Ω . Thus, W corresponds to a line L Ω ⊆ Q Ω = Q K ⊗ K Ω, and the zero locus (in −1 X Ω = X K ⊗ K Ω) of a nonzero section of W is equal to the closure of ψ Ω (L Ω ) ⊆ X Ω . def Now we are ready to interpret Lemma 16.1: Namely, I claim that Lemma 16.1 implies −1 (L Ω ) is deﬁned over K. that no irreducible component of the (closure of the) divisor ψ Ω 81 Indeed, if there were a divisor D ⊆ X K such that ψ Ω (D Ω ) ⊆ L Ω , then by spreading out Ω (respectively, L Ω ) to a smooth aﬃne curve C K over K (respectively, a family of lines in Q K parametrized by C K ), we obtain that for every closed point c ∈ C K , we have ψ(D) ⊆ L c . But since (by Lemma 16.1) L Ω is not deﬁned over K, it follows that for two suﬃciently generic closed points c, c ∈ C K , dim K (L c L c ) = 0, which implies that D is contained in the inverse image of a ﬁnite set of closed points of Q K . But we saw in the preceding paragraph that the inverse image via ψ of a closed point of Q K is of codimension ≥ 2. Since D is a divisor, this is absurd. This completes the proof of the claim stated at the beginning of this paragraph. Next, let us base-change U K and X K to K : this gives rise to U K and X K . The def two-dimensional subspace W ⊆ V K deﬁnes a rational map from X K to P K = P( W ). Let X K ⊆ X K be the complement of the indeterminacy locus of this rational map. Thus, the complement of X K in X K is of codimension ≥ 2, and, moreover, we obtain a dominant morphism (a “pencil”) ξ : X K → P K Let η P be the generic point of P K . Let η P be the spectrum of an algebraic closure of K(η P ) (the function ﬁeld of P K ). Let def F η P ⊆ X η P = X K × K η P be the divisor which is the zero locus of the section of L ⊗ K K(η P ) deﬁned by the generic point η P of P K . (Thus, F η P → η P is proper.) Put another way, F η P is the (closure in X η P of the) ﬁber (⊆ X η P ⊆ X η P ) of the morphism ξ over the generic point η P ∈ P K (whence the use of the letter “F ”). Since the pencil P K contains e 1 , which is a genericization of e 1 (i.e., e 1 is the specialization of e 1 at s 1 = t = 0), and the zero locus of e 1 is (geometrically) smooth and connected, it follows that F η P will also be geometrically smooth and connected over η P . Moreover, by the Lefshetz hyperplane theorem (see, e.g., [SGA2]), it follows (since dim K (X K ) ≥ 2) that the natural morphism π 1 (F η P ) → π 1 (X η P ) = π 1 (X K ) = π 1 (X K ) is surjective. Let def G η P = F η P × X ηP U η P = F η P × X K U K Since U K is a projective limit of dense open subschemes of X K , it thus follows that G η P is a projective limit of open subschemes of F η P . Moreover, since F η P (thought of as (the closure of) the ﬁber of the dominant morphism ξ over the generic point η P of P K ) contains (as a dense open subset) a scheme which is a projective limit of dense open subschemes of X K , we conclude that G η P is an integral, nonempty scheme. 82 At any rate, we get a natural morphism π 1 (G η P ) → π 1 (U η P ) = π 1 (U K ) = π 1 (U K ) Now we have the following important Lemma 16.2. This morphism π 1 (G η P ) → π 1 (U K ) is surjective. → U (of degree > 1), Proof. It suﬃces to take a ﬁnite, connected étale covering U K K pull it back to a covering G η P → G η P over G η P , and show that this pulled back covering η → G η is split. Now observe that since ξ is can never be split. Indeed, suppose that G P P dominant, it follows that F η P → X K and G η P → U K are dominant. In fact, F η P → X K and G η P → U K are even birational isomorphisms. Moreover, I claim that the divisors (in X K ) that were thrown out of X K to create F η P – let us call these divisors F -divisors – are diﬀerent from the divisors that were thrown out of X K to create U K – which we shall call U-divisors. Indeed, as divisors of X K = X K ⊗ K K , the U-divisors are all deﬁned over ﬁnite extensions of K, whereas the F -divisors (which are just ﬁbers of ξ over closed points of P K ) are, by Lemma 16.1 (see also the interpretation of Lemma 16.1 in the two paragraphs following the proof of Lemma 16.1), never deﬁned over ﬁnite extensions of K. This proves the claim. Next, let us observe that the morphism F η P → F η P is ramiﬁed only over F -divisors. Thus, it is unramiﬁed over U-divisors. Since G η P → F η P is birational, the fact that η → G η splits (hence, in particular, extends to a ﬁnite étale covering over F η ) thus G P P P → U arises → U is unramiﬁed over the U-divisors, i.e., U implies that the original U K K K K → X . (Here we use “purity of the branch locus” for regular local from a covering X K K rings – see, e.g., [SGA2], Exposé X, p. 118, Théorème 3.4.) But then the fact that the pull-back F η P → F η P of this covering to F η P does not split follows from the surjectivity observed immediately before the statement of this Lemma. Thus, since G η P → F η P is η → G η cannot split either. This contradiction completes birational, it follows that G P P the proof of the Lemma. Now let us assume that we are given a surjective continuous homomorphism over Γ K θ : Π U K → Π Y K def where Y K is a proper hyperbolic curve over K. Let Y η P = Y K × K η P . Thus, Y η P is a proper hyperbolic curve over η P . Suppose, moreover, that we know that morphisms like θ necessarily arise geometrically (i.e., from a dominant morphism U K → Y K ) for U K of dimension < n. (We shall refer to this assumption as the “Induction Hypothesis.” 83 Note that by Corollary 15.5, we already know that this induction hypothesis is true for dim K (U K ) = 1.) Then the morphism Π G η → Π Y K P obtained by composing (“π 1 of”) the natural morphism G η P → U K with θ lies over Γ η P → Γ K , and by Lemma 16.2, is such that it induces a surjection between the (maximal pro- p quotients of the) respective geometric fundamental groups. Moreover, this morphism naturally deﬁnes a morphism θ G : Π G η → Π Y η P P (since Π Y η is the ﬁbered product of Π Y K and Γ η P over Γ K ). Thus, to summarize, θ G is P a surjective continuous homomorphism over Γ η P . Let U η G P be the generic point of G η P . Thus, by composing (“π 1 of”) U η G P → G η P with θ G , we obtain a continuous surjective homomorphism θ U G : Π U η G → Π Y η P P over η P . Moreover, the dimension of U η G P over η P is < n. Thus, by the induction hypothesis, it follows that θ U G arises geometrically, from some unique dominant morphism U η G P → Y η P . Projecting to Y K then gives a dominant morphism U η G P → Y K . We would like to observe that this morphism U η G P → Y K extends to U K . Indeed, to see this, observe that U η G P and U K are both projective limits of open subsets of the projective K -variety X K . Moreover, U K can be written as the projective limit of K - smooth open subsets of X K . Thus, the fact that we get a morphism U K → Y K follows from the following Lemma 16.3. Let L be a ﬁeld; Y L be a proper, smooth, geometrically connected curve of nonzero genus over L; and Z L be a smooth L-variety. Suppose that we have a rational map φ from Z L to Y L . Then φ is deﬁned over all of Z L . Proof. This Lemma is a well-known consequence of the classical theory of the Albanese variety (see, e.g., [Lang]): Namely, let U L ⊆ Z L be a (nonempty) open over which φ is deﬁned. Assume (without loss of generality) that Y L (L) = ∅. Let A L be the Jacobian of Y L , and let B L be its dual. Then a point of Y L (L) deﬁnes an embedding Y L → A L which we can compose with φ to obtain a rational map φ A from Z L to A L . It suﬃces to show 84 that φ A extends to a morphism on Z L . But note that the portion of φ A which already is a morphism (from U L to A L ) deﬁnes (by pulling back the Poincaré bundle on A L × L B L ) a line bundle on U L × L B L . Since Z L × L B L is a regular scheme, it follows that this line bundle extends to Z L × L B L . Then taking the classifying morphism associated to this extended line bundle gives a morphism Z L → A L , as desired. Let us review what we have done so far. We started with a continuous surjective homomorphism θ : Π U K → Π Y K over Γ K . Moreover, we have shown that if we base-change θ from K to K to obtain θ K : Π U K → Π Y K then θ K arises geometrically from some dominant U K → Y K . Note, moreover, that this morphism U K → Y K is the unique morphism that gives rise to (the geometric portion of) θ. (This follows, for instance, from the inductive hypothesis on n. Moreover, this uniqueness also holds, of course, over any ﬁnite extension of K .) Thus, if we specialize the indeterminate t ∈ K to some element of a ﬁnite extension L of K that is suﬃciently generic so that U K → Y K specializes to U L → Y L , and then base-change this U L → Y L back up to a morphism U L → Y L (where L is the composite of K and L), then this morphism U L → Y L coincides with the morphism obtained by base-changing the original U K → Y K via K ⊆ L . But this means that the original U K → Y K is, in fact, deﬁned over L. Finally, since K and L are linearly disjoint over K, it follows that the original U K → Y K is deﬁned over K. Thus, we get a morphism U K → Y K which clearly induces the original θ (since, for instance, Π U K → Π U K is surjective). Let us step back now and take stock of what we have done so far in this Section. We started with U K , the spectrum of a function ﬁeld over K, and a proper hyperbolic curve Y K over K. Then given any surjective continuous homomorphism θ : Π U K → Π Y K over Γ K , we showed that θ necessarily arises geometrically. Now let us pause for a deﬁni- tion: Deﬁnition 16.4. Let Q K be a K-scheme. We shall call Q K a smooth pro-variety if it is the projective limit of a projective system of smooth (geometrically connected) varieties over K such that the transition morphisms are all birational. Now Lemma 16.3, plus the techniques of the proofs of Theorem 14.1 and Corollary 14.2 show that we have, in fact, proven the following (our ﬁrst main theorem – Theorem A in the Introduction): 85 Theorem 16.5. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let X K (respectively, Y K ) be a smooth pro-variety (respectively, hyperbolic pro-curve) over K. Let Hom dom K (X K , Y K ) open the set of dominant K-morphisms from X K to Y K . Let Hom Γ K (Π X K , Π Y K ) be the set of open, continuous group homomorphisms Π X K → Π Y K over Γ K , considered up to composi- tion with an inner automorphism arising from Δ Y . Then the natural map open Hom dom K (X K , Y K ) → Hom Γ K (Π X K , Π Y K ) is bijective. Remark. Finally, we make the following important observation: Note that given any pro-p result such as Theorem 16.5, one can always immediately derive a corresponding proﬁnite result from it. (Here by “corresponding proﬁnite result,” we mean the same result, except that “Π” (respectively, “Δ”) is replaced by “Π prf ” (respectively, “Δ prf ”).) Indeed, suppose that K, X K , and Y K are as in the statement of Theorem 16.5, and let prf θ : Π prf X K = π 1 (X K ) → Π Y K = π 1 (Y K ) be an open homomorphism over Γ K . Then note that θ immediately induces an open homomorphism θ p : Π X K → Π Y K . Applying Theorem 16.5 shows that θ p arises from some φ : X K → Y K . Thus, it remains only to show that the morphism induced by φ on fundamental groups coincides with θ. But this follows from the argument preceding Theorem 14.1: namely, we consider an arbitrary ﬁnite étale covering Y K → Y K . Pulling this covering back via θ gives a ﬁnite étale covering X K → X K . Moreover, θ induces a morphism between the full proﬁnite fundamental groups of X K and Y K . Next, observe that this morphism gives us a φ : X K → Y K (by the same argument as that used to construct φ from θ) which (by naturality of the constructions involved) lies over φ. Thus, if we continue this procedure for arbitrary ﬁnite étale coverings, the well-known correspondence between fundamental groups and categories of étale coverings shows that the morphism induced by φ on fundamental groups coincides with θ up to composition with an inner automorphism induced by an element of Δ prf Y . This completes the proof of the “proﬁnite analogue of Theorem 16.5.” Section 17: Maps Between Higher-Dimensional Function Fields Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let L and M be function ﬁelds (of arbitrary dimension) over K. (Thus, in particular, we assume here that K is algebraically closed in L and M). We denote by Γ L and Γ M the absolute Galois groups of L and 86 M, respectively. In this Section, we would like to show how to derive a result (Theorem 17.1, which is stated as Theorem B in the Introduction) like Theorem 16.5 for morphisms between L and M (over K). Theorem 17.1. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let L and M be func- tion ﬁelds of arbitrary dimension over K. Let Hom K (Spec(L), Spec(M)) be the set of K-morphisms from M to L. Let Hom open Γ K (Γ L , Γ M ) be the set of open, continuous group homomorphisms Γ L → Γ M over Γ K , considered up to composition with an inner automor- phism arising from Ker(Γ M → Γ K ). Then the natural map Hom K (Spec(L), Spec(M)) → Hom open Γ K (Γ L , Γ M ) is bijective. Proof. First, recall (cf. the Remark following Theorem 16.5) that any pro-p result such as Theorem 16.5 always implies a corresponding proﬁnite result. Now we use induction on the transcendence degree – which we shall henceforth denote by dim K (M) – of M over K. When dim K (M) = 0, the result is vacuous. When dim K (M) = 1, the result follows from the proﬁnite version of Theorem 16.5; thus, we may assume that dim K (M) > 1. Now suppose that we know Theorem 17.1 to be true for maps to functions ﬁelds of transcendence degree < dim K (M). Suppose that we are given an open continuous homomorphism θ : Γ L → Γ M over Γ K . Observe that just as previously (e.g., in the proof of Theorem 14.1), we can assume without loss of generality that θ is surjective. Next, observe that there exists a function ﬁeld P ⊆ M such that 0 < dim K (P ) < dim K (M). Moreover, we may assume that P is algebraically closed in M. Thus, we get a surjection Γ M → Γ P . Composing θ with this surjection, we obtain a surjection θ P : Γ L → Γ P , which, by the induction hypothesis on dim K (M), we know arises geo- metrically from some P → L (over K). Moreover, since θ P is surjective, it follows that P is algebraically closed in L. Thus, we may regard L and M as function ﬁelds over P . Moreover, dim P (M) < dim K (M). Thus, since θ : Γ L → Γ M is (by the deﬁnition of θ P ) a morphism over Γ P , it follows from the induction hypothesis on dim K (M) that θ arises from some M → L over P (hence also over K). This completes the proof of the Theorem. Remark. Note that in this case, we needed to work with proﬁnite (not pro-p) fundamental groups because the operation of taking the maximal pro-p quotient is not well-behaved with respect to ﬁbrations: i.e., if we had replaced Ker(Γ M → Γ K ) with its maximal pro-p quotient, we would run into trouble because it is not clear that the maximal pro-p quotient of Ker(Γ M → Γ P ) injects into the maximal pro-p quotient of Ker(Γ M → Γ K ). 87 Also, we needed to work with function ﬁelds (as opposed to varieties) because, if, for instance, Spec(L) and Spec(P ) had been varieties, the fact that the geometric (over K) fundamental group of Spec(L) surjects onto that of Spec(P ) does not necessarily imply that the morphism Spec(L) → Spec(P ) has geometrically connected ﬁbers. Moreover, if the generic geometric ﬁber of Spec(L) → Spec(P ) has several distinct connected components, it is not necessarily the case that the fundamental group of any of these connected components surjects onto the geometric (over P ) fundamental group of Spec(M). Indeed, by replacing Spec(M) by a smooth projective variety Y K (of dimension ≥ 3), and Spec(L) by the result – call it X K – of cutting this variety with a generic hyperplane section, then the inclusion X K → Y K induces an isomorphism on geometric fundamental groups despite the fact that X K is not isomorphic to Y K . Moreover, note that such a counterexample to a “variety version” of Theorem 17.1 exists even if Y K is “hyperbolic” (say, a product of proper hyperbolic curves), in which case, one would expect X K (at least if the ample line bundle used to cut Y K to form X K has suﬃciently high degree) to be “quite hyperbolic,” as well. Section 18: Truncated Fundamental Groups def If Δ is a topological group, let us introduce the following notation: Δ{0} = Δ; for def i ≥ 1, Δ{i} = [Δ{i−1}, Δ{i−1}]. Also, let us write “Π i ” (respectively, “Δ i ”) for Π/Δ{i} (respectively, Δ/Δ{i}). The purpose of this Section is to observe that much of the theory of this paper continues to hold to a large extent even when we consider morphisms not between the full Π’s, but between certain truncated versions of the Π’s. Our ﬁrst main result is the following: Theorem 18.1. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let X K be a smooth variety over K. Let Y K be a hyperbolic curve over K. Let n ≥ 5. Then every continuous open homomorphism θ : Π nX K → Π nY K over Γ K induces a dominant morphism μ : X K → Y K whose induced morphism on fun- damental groups coincides (up to composition with an inner automorphism arising from n−3 Δ Y K ) with the morphism Δ n−3 X K → Δ Y K deﬁned by considering θ “modulo Δ{n − 3}.” Proof. First let us consider the case where: (i) K is a ﬁnite extension of Q p ; (ii) X K is a curve; (iii) Y K is a non-hyperelliptic proper hyperbolic curve; (iv) n = 3. Then note that the theory of Sections 1 through 5 manifestly only involves Δ 2 X K . It thus follows immediately (via the help of a technical lemma – Lemma 18.4 below – necessary in order to assure that Lemma 7.3 goes through in the truncated context) that to prove Proposition 7.4, we really only used Δ 2 Y η . Thus, in summary, as long as the covering Y η S → Y η S (in S 88 the notation of the discussion preceding Proposition 7.4) arises from an open subgroup of Π 1 Y K , we can apply Proposition 7.4 to Y η S to conclude that Y η S admits a line bundle of degree prime to p. Now we would like to conclude the “preservation of relations.” Note that Section 11 manifestly only involves the abelianization of the geometric fundamental group. More- over, Section 12 is just formal manipulation. In Section 13, the only thing from Sections 1 through 7 that is used is Proposition 7.4 (whose applicability under the present circum- stances was discussed in the preceding paragraph). Thus, by using Propositions 7.4 and 8.1, Lemma 8.2, and the theory of Section 9, we may conclude “preservation of relations” for the morphism H 0 (Y K , ω Y K /K ) → H 0 (X K , ω X K /K ) induced by Δ 1 X K → Δ 1 Y K . (Note, however, that this time, in the application of Section 9, we take for our tower of coverings “X L n ” not the coverings corresponding to the p-derivates of the whole geometric fundamen- tal group Δ, but rather the p-derivates of Δ 1 – which is enough to carry out the argument of Section 9.) Hence we get a morphism μ : X K → Y K . Now let us lift the hypothesis that n = 3. Then we obtain the following: If Y K → Y K is a covering arising from an open subgroup of Π n−3 Y K , and X K → X K is a connected component of the pull-back of this covering to X K via θ, then we get a natural morphism μ : X K → Y K . Moreover, the naturality of the construction of this morphism means that it always lies over the morphism X K → Y K . Thus, it follows by the usual argument (i.e., the one preceding Theorem 14.1) that the morphism induced by μ on fundamental groups is equal to θ modulo Δ{n − 3} (up to composition with an inner automorphism arising from Δ Y K ). So far we have not used that n ≥ 5. (In fact, we have only used that n ≥ 3.) The purpose of assuming that n ≥ 5 is to lift hypothesis (iii) (in the ﬁrst paragraph of this proof). Namely, given any hyperbolic curve Y K , it is elementary to show that there always exists a covering Y K → Y K deﬁned by an open subgroup of Π 2 Y K such that the compactiﬁcation Y K of Y K is hyperbolic and non-hyperelliptic, and such that Y K → Y K has arbitrarily large (speciﬁed) ramiﬁcation over all the points of Y K − Y K . Thus, we get a → Y K which descends to a map X K → Y K . Moreover, the fact that X K → X K map X K is étale, while Y K → Y K is ramiﬁed over the points of Y K − Y K (with ramiﬁcation indices arbitrarily large) implies that the map X K → Y K factors through Y K . Thus, we get a map X K → Y K , as desired. Then arguing as in the preceding paragraph completes the proof, albeit still under the assumptions (i) and (ii) (of the ﬁrst paragraph of the proof). The extension to the case of ﬁelds K that are subﬁelds of ﬁnitely generated extensions of Q p then follows via the same “specialization argument” as that employed previously in the nontruncated case (following Lemma 15.2). Thus, we can also lift assumption (i). Finally, the “cutting with a hyperplane argument” of Section 16 extends immediately to the truncated case. This allows us to lift assumption (ii), thus completing the proof of the Theorem. Theorem 18.2. Let K be a subﬁeld of a ﬁnitely generated ﬁeld extension of Q p . Let X K be a smooth pro-variety over K. Let Y K be a hyperbolic pro-curve over K. Let n 0 be 89 the minimum transcendence degree over Q p of all ﬁnitely generated ﬁeld extensions of Q p that contain K. Let n 0 be the transcendence degree over K of the function ﬁeld of X K . def Let n 0 = n 0 + 2(n 0 − 1) + 1. Let n ≥ 3n 0 + 5. Then every continuous open homomorphism θ : Π nX K → Π nY K over Γ K induces a dominant morphism μ : X K → Y K whose induced morphism on fundamental groups coincides (up to composition with an inner automorphism arising 0 0 from Δ Y K ) with the morphism Δ n−3−3n → Δ n−3−3n deﬁned by considering θ “mod- X K Y K ulo Δ{n − 3 − 3n 0 }.” Proof. The reason for the inclusion of the extra “3n 0 ” (i.e., a “price” of three steps for every additional transcendence degree that is used, plus an extra “tax” of three steps for allowing the preﬁx “pro”) is the following: in order to reduce to the situation discussed in the proof of Theorem 18.1, we need to show (in the present “pro” context) that inertia groups are annihilated. Moreover, to apply the ﬁrst “inertia annihilation argument” (at the beginning of Section 14), we needed to know that after one passes to some covering Y K → Y K arising from an open subgroup of Π n−3 Y K (i.e., we wish to apply the arguments of Theorem 18.1 for n − 3), one knows “preservation of relations” for Y K . Thus, already one needs some extra padding – to the tune of three steps (necessary, as we saw in the ﬁrst two paragraphs of the proof of Theorem 18.1, to derive “preservations of relations” for Y K ). (This accounts for the 1 in the deﬁnition of n 0 .) Moreover, each time one adds a transcendence degree, one needs to apply the “inertia annihilation argument” of the ﬁrst half of Section 15. Thus, by the same line of reasoning, one needs extra padding consisting of three steps. (This accounts for the n 0 in the deﬁnition of n 0 .) In Section 16, one uses not only the transcendence degrees inherent in K, but also 2(n 0 − 1) auxiliary transcendence degrees – here the “2 = 1 + 1” comes from the transcendence degree of “K ” over “K,” plus the transcendence degree of the pencil “ξ.” (This accounts for the 2(n 0 − 1) in the deﬁnition of n 0 .) Finally, we remark that although in Sections 14, 15, and 16, we assumed that the morphism of fundamental groups “θ” was surjective, it is easy to see that this assumption is merely cosmetic, i.e., is inessential and serves only to simplify the discussion. This completes the proof of the Theorem. Remark. Thus, the essential diﬀerence between Theorems 18.1 and 18.2 is that in Theo- rem 18.2, we allow “pro-objects,” at the cost of having to apply “annihilation of inertia” arguments, which require us to use larger quotients of Δ X K , Δ Y K (i.e., each application of annihilation of inertia costs three units of “n”). Remark. We do not mean to pretend that the estimates in the above two Theorems (e.g., the “5’s,” “n 0 ,” etc.) are the best possible. Especially if one is willing to add hypotheses 90 to Y K , it should not be so diﬃcult to improve these estimates. The point of the above two Theorems is simply to illustrate the principle involved. We conclude this Section with two technical lemmas that were used in the proofs of the above two Theorems. Lemma 18.3. Let X K be a proper hyperbolic curve over a ﬁeld K of characteristic zero. Let Δ X be (as usual) the maximal pro-p quotient of its geometric fundamental group. Let def Ξ X = Δ X /[Δ X , [Δ X , Δ X ]]. Then the natural morphism H 2 (Ξ X , Z p (1)) Γ K → H 2 (Δ X , Z p (1)) Γ K (induced by the quotient map Δ X → Ξ X ) is surjective. Here, the superscripted “Γ K ” denotes “the submodule of Γ K -invariants.” Proof. Let us ﬁrst consider the morphism H 2 (Ξ X , Z p (1)) → H 2 (Δ X , Z p (1)) To do this, we shall use the Hochschild-Serre spectral sequence for the quotient Ξ X → H X (where, as usual, we write H X for the abelianization of Δ X ). Write Ψ X ⊆ Ξ X for the kernel of Ξ X → H X . Thus, Ψ X may be identiﬁed with a certain well-understood (cf. Lemma 3.1) quotient of ∧ 2 H X . Consideration of the E 2 ·· -term of this spectral sequence shows that there is a natural Γ K -equivariant injection of the cokernel of the natural morphism (induced by the quotient ∧ 2 H X → Ψ X ) H 1 (Ψ X , Z p (1)) = Hom Z p (Ψ X , Z p (1)) → H 2 (H X , Z p (1)) = Hom Z p (∧ 2 H X , Z p (1)) into H 2 (Ξ X , Z p (1)). (In other words, the above morphism is the diﬀerential (from E 2 0,1 to E 2 2,0 ) of the “E 2 pq = H p (H X , H q (Ψ X , Z p (1)))”-term of the spectral sequence.) Moreover, it follows from Lemma 3.1 that the cokernel of this natural morphism is precisely the quotient of H 2 (H X , Z p (1)) = (∧ 2 H 1 (H X , Z p ))(1) given by the (surjective) cup-product map (∧ 2 H 1 (H X , Z p ))(1) → H 2 (Δ X , Z p )(1). Thus, in summary, we have a Γ K -equivariant diagram: H 2 (Ξ X , Z p (1)) ⊇ Image(H 2 (H X , Z p (1))) → H 2 (Δ X , Z p (1)) in which the arrow is bijective. This implies the assertion stated in the Lemma. 91 Lemma 18.4. Let K, X K , Δ X , and Ξ X be as in Lemma 18.3. Since the kernel of Δ X → Ξ X is a normal subgroup not only of Δ X , but also of Π X K , write Π X K → Π Ξ X K for the quotient of Π X K by this subgroup. Then the natural morphisms 2 H 2 (Π Ξ X K , Z p (1)) → H (Π X K , Z p (1)) and 2 H 2 (Π X K × Γ K Π Ξ X K , Z p (1)) → H (Π X K × Γ K Π X K = Π X K × K X K , Z p (1)) are surjective. In particular, if η ∈ H 2 (Π X K × K X K , Z p (1)) denotes the ﬁrst Chern class of the diagonal in X K × K X K , then η lies in the image of H 2 (Π X K × Γ K Π Ξ X K , Z p (1)). Proof. The assertions of this Lemma follow by considering the consequences of the sur- jectivity assertion of Lemma 18.3 for the Hochschild-Serre spectral sequences associated Ξ to Π Ξ X K → Γ K ; Π X K → Γ K ; Π X K × Γ K Π X K → Γ K ; and Π X K × Γ K Π X K → Γ K . Section 19: Injectivity Result In this Section, we prove the following “pro-p injectivity part of the so-called Section Conjecture”: Theorem 19.1. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let X K be a hyperbolic curve over K. Let X K (K) be the set of K-valued points of X K . Let Sect(Γ K , Π X K ) be the set of sections Γ K → Π X K of Π X K → Γ K , considered up to composition with an inner automorphism arising from Δ X K . Then the natural map X K (K) → Sect(Γ K , Π X K ) is injective. Remark. The so-called “Section Conjecture” states that (for instance when X K is proper) the natural map in Theorem 19.1 is bijective. (When X K is aﬃne, the statement must be modiﬁed slightly.) At the time of writing (July 1996), the Section Conjecture has not yet been proven. Proof. (of Theorem 19.1) Let H X be the abelianization of Δ X . First, note that by replacing X K by some ﬁnite Galois étale covering of X K of p-power order, we may assume 92 that the compactiﬁcation X K of X K is itself hyperbolic and non-hyperelliptic. Let r be the number of points in (X K − X K )(K). Let Z X be the complement of the diagonal in X K × K X K . By taking the second projection Z X → X K , we may regard Z X as a family (parametrized by X K ) of smooth hyperbolic curves obtained by removing r + 1 distinct points from some compactiﬁcation. Let η X be the generic point of X K . Let η X be the spectrum of some algebraic closure def of K(η X ). Let Z η X = Z X × X η X . Then (since Z X → X K is a family of smooth hyperbolic curves obtained by removing precisely r + 1 distinct points from some compactiﬁcation) we obtain an exterior Galois representation ρ Z : Π prf X K → Out(Δ Z η ) X Now I claim that ρ Z factors through Π X K . Indeed, to see this, note that if H Z ηX is the abelianization of Δ Z η , then since Δ Z η is a pro-p group, the kernel of X X Ξ : Out(Δ Z η ) → Aut(H Z η X ) X is itself a pro-p group. Thus, it suﬃces to show that Ξ◦ρ Z maps Δ prf X into a pro-p subgroup of Aut(H Z ηX ). But now observe that one has a Galois-equivariant surjection H Z ηX → H X whose kernel is either 0 or Z p (1) (depending on whether r = 0 or r > 0). Moreover, the actions of Δ prf X on H X and Z p (1) are trivial. Thus, it follows immediately that the prf Δ X -action on H Z ηX is by unipotent matrices, so Ξ ◦ ρ Z maps Δ prf X into a pro-p subgroup of Aut(M Z ), as desired. This completes the proof of the claim. Thus, we have a representation ρ Z : Π X K → Out(Δ Z η ) X Now suppose that we have two points α, β ∈ X K (K) that induce the same element φ ∈ Sect(Γ K , Π X K ). Let Z α (respectively, Z β ) be the pull-back of Z X → X via α (respectively, β). Thus, Z α and Z β are hyperbolic curves over K. Moreover, since the action of Γ K on Δ Z α (respectively, Δ Z β ) is determined by composing φ with ρ Z , it follows that there exists some isomorphism ψ : Δ Z α ∼ = Δ Z β such that (i) ψ is compatible with the respective outer Γ K -actions; (ii) ψ preserves and induces the identity between the quotients Δ Z α → H X , Δ Z β → H X . Thus, by Theorem 16.5 (in fact, really, all we need is Theorem A of [Mzk2]), it follows that ψ arises from some isomorphism Z α ∼ = Z β which is compatible with and induces the identity on the inclusions Z α → X K , Z β → X K . But this clearly implies that α = β, thus completing the proof of the Theorem. Remark. Note that in fact, in the proof of Theorem 19.1, we really only used the weaker results of [Mzk2] – i.e., we did not need to use Theorem A of the present paper. On the 93 other hand, at the time that [Mzk2] was written, the author was unaware of Theorem 19.1, which is the primary reason that Theorem 19.1 did not appear in [Mzk2]. Note also that although Theorem 19.1 implies a corresponding proﬁnite result, the proﬁnite result can be proven much more easily, by using the “Kummer exact sequence” for the Jacobian of X K . Finally, it should be remarked that the argument employed in the proof of Theorem 19.1 (as well as the Kummer sequence argument just mentioned) have been well-known for some time (see, e.g., [Naka2]). The main reason that Theorem 19.1 was included in this paper was that the author just wanted to make explicit that a pro-p injectivity result (such as Theorem 19.1) could now be proven. APPENDIX: A Grothendieck Conjecture-Type Result for Certain Hyperbolic Surfaces Section a0: Introduction In this Appendix, we show how Theorem A (cf. the Introduction) can be used to prove a Grothendieck Conjecture-type result for certain types of surfaces. The surfaces considered are families of (smooth) hyperbolic curves that are parametrized by hyperbolic curves (cf. Deﬁnition a2.1). We call such surfaces hyperbolically ﬁbred. Our notation is similar to that of the rest of the paper, except that since here we consider only proﬁnite (i.e., not pro-p) fundamental groups, in this Appendix, we will write Π X K for the proﬁnite fundamental group of X K : def Notation: If K is a ﬁeld and X K is a K-scheme, we denote by Π X K = π 1 (X K ) the fundamental group of X K (for some choice of base-point), and by Γ K the absolute Galois group of K. Then we have a natural morphism Π X K → Γ K whose kernel Δ X ⊆ Π X K is the geometric fundamental group π 1 (X K ⊗ K K) (where K is an algebraic closure of K). Our main theorem is the following: Theorem D. Let K be sub-p-adic (cf. Deﬁnition 15.4 (i)). Let X K and Y K be hyperbolically ﬁbred surfaces over K. Let Isom K (X K , Y K ) be the set of K-isomorphisms (in the category of K-schemes) between X K and Y K . Let Isom Γ K (Π X K , Π Y K ) be the set of continuous group isomorphisms Π X K → Π Y K over Γ K , considered up to composition with an inner automorphism arising from Δ Y . Then the natural map Isom K (X K , Y K ) → Isom Γ K (Π X K , Π Y K ) 94 is bijective. This Theorem is given as Theorem a2.4 in the text. We remark that: (1) Theorem D above is (modulo a certain technical result – Lemma a1.1) essentially derived from Theorem A. Note that one needs the full power of the “Hom version” of the Grothendieck Conjecture for hyperbolic curves (i.e., Theorem A) in order to prove an “Isom-type result” for surfaces (i.e., Theorem D above). That is to say, the “Isom version” of Theorem A of the present paper – i.e., Theorem A of [Mzk2] – is not suﬃcient to prove Theorem D above. Moreover, even with the “Hom version” of the Grothendieck Conjecture for hyperbolic curves (i.e., Theorem A), at the present time, I am unable to prove a “Hom version” of Theorem D. (2) One might ask whether Theorem D can be extended to hyperbolically ﬁbered varieties (i.e., varieties obtained as successive ﬁbrations of hy- perbolic curves) of higher dimension. The problem is that just as we needed (see Remark (1)) the “Hom version” in dimension one to prove an “Isom-type result” in dimension two, we would need a “Hom-type result” in dimension two – which is currently not available – in order to prove an “Isom-type result” in, say, dimension three. Thus, at the present time, we are unable to advance beyond dimension two. (3) Since Theorem A is valid in the pro-p case as well, one might ask why one cannot prove a pro-p version of Theorem D above. The problem is that the process of passing to the maximal pro-p quotient is not well-behaved with respect to ﬁbrations. That is to say, if X K → X K is a family of hyperbolic curves parametrized by a hyperbolic curve (as in Deﬁnition a2.1), then the maximal pro-p quotient of the fundamental group of the geometric generic ﬁber of X K → X K does not map injectively (in general) to the maximal pro-p quotient of Δ X . Thus, any attempt to prove a pro-p version of Theorem D by means of the techniques employed here would result in a rather unnatural theorem. (4) Since Theorem A admits various truncated versions (cf. the Introduc- tion) as well, one might ask why one cannot prove a truncated version of Theorem D. The problem here is the same as the problem that arises when one tries to prove a pro-p result: i.e., truncating is not well-behaved with respect to ﬁbrations. (5) To a slight extent, the content of Theorem D above overlaps with recent results of H. Nakamura and N. Takao (see Theorem A and Corollary B of [NT]). 95 Section a1: A Key Lemma In this Section, we prove a simple technical lemma which will be the key technical ingredient that allows us to extend Theorem A to the case of surfaces. Let K be an algebraically closed ﬁeld of characteristic zero. Let X and Y be hyperbolic curves over K (cf. Section 0 for a deﬁnition of this term). Lemma a1.1. Let φ : X → Y be a ﬁnite K-morphism. Let ψ : Π X K → Π Y K be the induced morphism on fundamental groups. Suppose that Ker(ψ) is topologically ﬁnitely generated. Then φ is étale. Proof. By replacing Y by the ﬁnite étale covering of Y corresponding to Im(ψ) ⊆ Π Y K , we may assume that ψ is surjective. Under this assumption, φ is étale if and only if φ is an isomorphism. Thus, it suﬃces to assume that: (1) Ker(φ) is topologically ﬁnitely generated; and (2) φ is not an isomorphism (hence has degree > 1) and derive a contradiction. def Let Y → Y be a ﬁnite étale covering Y , where Y is connected. Let X = X × Y Y . Then X → X is ﬁnite étale, and it follows from the assumption that ψ is surjective that X is connected. Since Y is hyperbolic, it follows that for a suitable choice of Y → Y , Y will have genus ≥ 2. Thus, by replacing our original X → Y by X → Y , we may assume that X and Y have genus ≥ 2. Since X → Y is assumed to have degree > 1, it follows that g X (the genus of X) is strictly greater than g Y (the genus of Y ). Let Y → Y be a ﬁnite étale covering of degree d such that Y is connected, and Y → Y extends to an étale covering over some compactiﬁcation of Y . Then the Riemann-Hurwitz formula implies that, as Y → Y varies, g Y (which we think of as a function of d) is equal to d(g Y − 1) + 1. On the other hand, def if X = X × Y Y , then (again by the Riemann-Hurwitz formula) g X = d(g X − 1) + 1. Thus, it follows that as d → ∞, the diﬀerence g X − g Y → ∞. Now let H X denote the ﬁrst homology group of the compactiﬁcation of X with coeﬃcients in Z p (for some prime number p which will be ﬁxed throughout the discussion). Then ψ induces a surjection H ψ : H X → H Y . Thus, Ker(H ψ ) is a free Z p -module of rank 2(g X − g Y ). On the other hand, any topological generators of Ker(ψ) clearly deﬁne a set of Z p -generators of Ker(H ψ ). But this implies that rank Z p (Ker(H ψ )) is bounded, independently of Y → Y , which contradicts the fact that rank Z p (Ker(H ψ )) = 2(g X − g Y ) → ∞ as d → ∞. This contradiction completes the proof. 96 Section a2: The Main Theorem Let K be a ﬁeld of characteristic zero. Let X K be a surface over K, by which we mean that X K is a smooth (geometrically connected) variety over K of dimension two. Deﬁnition a2.1. We shall say that X K is a hyperbolically ﬁbred surface if the following condition holds: There exists a hyperbolic curve X K over K, together with a smooth, proper, connected morphism X K → X K of relative dimension one such that X K embeds as an open subvariety of X K satisfying the following conditions: (i) X K − X K is a divisor ; (ii) the geometric ﬁbers of X K ⊆ X K → X K are hyperbolic in X K which is étale over X K curves. If X K is a hyperbolically ﬁbred surface, then we shall refer to X K → X K (as above) as a parametrizing morphism for X K . Note that given a hyperbolically ﬁbered surface X K , in general, there can exist more than is a parametrizing morphism for one parametrizing morphism for X K . If X K → X K X K , and F Ω ⊆ X K is a ﬁber of this morphism (over some point of X K (Ω) valued in an algebraically closed ﬁeld Ω), then we have an exact sequence of fundamental groups: 1 → Π F Ω → Π X K → Π X → 1 K are hyperbolic curves, it follows that Π F Ω and Π X (hence Note that since F Ω and X K K also Π X K ) are topologically ﬁnitely generated. Now let K be sub-p-adic (cf. Deﬁnition 15.4 (i)) (for some prime number p), and let X K and Y K be hyperbolically ﬁbred surfaces over K. Let φ : Π X K → Π Y K be a continuous group isomorphism over Γ K . We would like to show that φ arises geomet- rically. Let ζ Y : Y K → Y K be a parametrizing morphism for Y K . Then by composing φ with π 1 (ζ Y ), we obtain a continuous surjection Π X K → Π Y . By Theorem A, it follows K that this surjection arises geometrically, from some morphism X K → Y K . Thus, we may regard X K and Y K as objects over Y K . Let Z K be the normalization of Y K in X K . Thus, the morphism X K → Y K factors through δ : Z K → Y K . Observe that Z K is a smooth, geometrically connected (since X K is geometrically connected over K) curve over K, and that Z K → Y K is ﬁnite. (In particular, since Y K is hyperbolic, it follows that Z K is also hyperbolic.) Moreover, it follows from the deﬁnition of Z K that the morphism Π X K → Π Z K induced by X K → Z K is surjective. This implies that Ker(π(δ)) is a quotient of Ker(Π X K → Π Y ) ∼ = Ker(Π Y K → Π Y ), which (by the deﬁnition of a hyperbolically ﬁbred K K 97 surface) is topologically ﬁnitely generated. Thus, it follows that Ker(π(δ)) is topologically ﬁnitely generated. But then Lemma a1.1 implies that Z K → Y K is étale. On the other hand, since Π X K → Π Y is surjective, we thus see that Z K → Y K must be an isomorphism. K In particular, we thus conclude that the generic ﬁber of X K → Y K is smooth, geometrically connected, and of dimension one. In fact, the argument of the preceding paragraph can be applied more generally to coverings of X K , as follows: Let Y K → Y K be a ﬁnite étale covering such that Y K is geometrically connected over K. Let Y K → Y K × Y K Y K be a ﬁnite étale covering such that K → X K be the covering corresponding (via Y K → Y K is geometrically connected. Let X K → X K factors through X K → X K × Y Y . Then I claim that φ) to Y K → Y K . Thus, X K K K → Y is geometrically connected. Indeed, this follows by the same argument as that X K employed in the preceding paragraph: Namely, we simply observe that Π → Π is K K X Y isomorphic to Π → Π , which is surjective with topologically ﬁnitely generated kernel. K K Y Y This proves the claim. Let L be the function ﬁeld of Y K ; let L be its algebraic closure. Note that the natural map Spec(L) → Y K factors through Y K . Moreover, the above claim K × Spec(L) K → Y via Spec(L) → Y , the resulting X implies that if we base-change X K K K Y is connected. Now let us reinterpret the conclusions of the preceding paragraph in terms of funda- def def mental groups. Let η Y be the generic point of Y K . Let Y η Y = Y K × Y K η Y ; X η Y = X K × Y K η Y . Then what we did in the preceding paragraph implies precisely that the morphism Ker(π 1 (X η Y ) → π 1 (η Y )) → Ker(π 1 (Y η Y ) → π 1 (η Y )) = Ker(Π Y K → Π Y ) K (induced by φ) is surjective. Thus, we see that by composing the natural surjection of π 1 (X η Y ) onto Π X K with φ, we obtain a continuous surjective group homomorphism φ η Y : Π X η → Π Y η Y Y over Γ η Y (where we regard X η Y and Y η Y as curves over η Y ). Now we would like to apply Theorem A again. This Theorem tells us that φ η Y arises geometrically from some η Y -morphism X η Y → Y η Y . Remark. Note that in the argument of the preceding three paragraphs, it was absolutely essential to invoke Lemma a1.1 before we could apply Theorem A. In fact, if the variety “on the left” (i.e., in this case, X η Y ) is not geometrically connected over the base ﬁeld (i.e., in this case, L), then it is not diﬃcult to see that Theorem A does not hold. Indeed, to construct such a counterexample, suppose that Y K is proper, and let H K ⊆ Y K be a 98 hyperplane section (with respect to some projective embedding of Y K ) such that H K is smooth and geometrically connected over K, and π 1 (H K ) → π 1 (Y K ) is surjective. (Such an H K exists by the Lefshetz hyperplane theorem.) Write η H for the generic point of H K , and take for our “X η Y ” any hyperbolic curve over η H . Then note that (if H K is suﬃciently generic so that every ﬁber of H K → Y K contains at least one point at which H K → Y K is étale, then) the induced morphism π 1 (η H ) → π 1 (Y η Y ) is also surjective, so we get a surjective morphism π 1 (X η Y ) → π 1 (Y η Y ) (by composing the (necessarily surjective) structure morphism π 1 (X η Y ) → π 1 (η H ) with π 1 (η H ) → π 1 (Y η Y )). But this surjective morphism does not arise from a dominant morphism X η Y → Y η Y . This completes the portion of the proof which is an application of (the nontrivial, sur- jectivity part of) Theorem A. The remainder of the proof will consists of using elementary algebraic geometry to show that the morphism X η Y → Y η Y constructed above extends to a morphism X K → Y K . Lemma a2.2. The morphism X η Y → Y η Y above extends to a morphism X K → Y K . Proof. First, observe that there exists a ﬁnite étale covering V K → Y K with the following properties: (1) V K is a hyperbolically ﬁbered surface that admits a parametrizing mor- phism V K → V K that ﬁts into a commutative diagram: V K ⏐ ⏐ −→ V K ⏐ ⏐ Y K −→ Y K (2) The ﬁbers of V K → V K have genus ≥ 2. Let U K → X K be the ﬁnite étale covering that corresponds (via φ) to V K → Y K . Let V K → V K be the family of proper hyperbolic curves that compactiﬁes V K → V K (as in Deﬁnition a2.1). Write η V for the generic point of V K . Note that there exists a natural map U K → V K covering X K → Y K , and that both U K and V K are geometrically connected over K → Y is geometrically connected). η V (cf. the argument used above to show that X K Thus, we may form (geometrically connected) η V -curves U η V , V η V . Moreover, by the deﬁnition of U K , we have a commutative diagram U η V ⏐ ⏐ −→ V η V ⏐ ⏐ X η Y −→ Y η Y 99 −→ V η V Thus, in particular, we obtain a morphism U η V → V η V (over η V ). I claim that this morphism extends to a morphism U K → V K . Indeed, by the elementary theory of surfaces – “elimination of indeterminacy” (see, e.g., Theorem 5.5 of Chapter V of [Harts]) – it follows K → U K (obtained by successively blowing up smooth points) such that there exists some U that the birational transformation from U K to V K deﬁned by U η V → V η V extends to a K necessarily maps K → V K . On the other hand, any exceptional curve in U morphism U into a ﬁber of V K → V K (this follows since U K → V K is already a morphism). Thus, any K maps quasi-ﬁnitely into a ﬁber of V K → V (which will always be a exceptional P 1 in U K smooth, proper, hyperbolic curve) – which is clearly absurd. This contradiction completes the proof of the claim. Thus, we have a morphism U K → V K . Let Y K → Y K compactify Y K → Y K (as in Deﬁnition a2.1). Then clearly V K → Y K extends to a ﬁnite (in general, ramiﬁed) morphism V K → Y K . Thus, to summarize, we have a rational map from X K to Y K which is covered by a morphism U K → V K , where U K (respectively, V K ) is ﬁnite over X K (respectively, Y K ). By elementary algebraic geometry, this implies that we get a morphism X K → Y K (covered by U K → V K ). To complete the proof of the Lemma, we must verify that this morphism X K → Y K factors through Y K . To do this, we simply choose V K → Y K in the above argument such that V K is ramiﬁed, with very large (say, compared to the degree of X K → Y K ) ramiﬁcation index, over all of the divisor Y K − Y K . (Note that such coverings exist, by the exact sequence of fundamental groups following Deﬁnition a2.1, plus the well-known fact from topology that the fundamental group of a compact surface with ﬁnitely many punctures has coverings with arbitrarily large ramiﬁcation index at those punctures.) Then since X K → Y K is covered by U K → V K , and U K → X K is ﬁnite étale, we see that we obtain a contradiction, unless X K → Y K maps into Y K . Thus, we conclude that we have a morphism X K → Y K , as desired. Let us summarize what we have done so far. We started with a continuous group isomorphism φ : Π X K → Π Y K over Γ K . From φ, we constructed a K-morphism X K → Y K , which we denote by α. Since φ is invertible, we thus see that we have also constructed a K-morphism β : Y K → X K from φ −1 . Since everything we have been doing is functorial, it follows that π 1 (β ◦ α) is the identity on Π X K (up to composition with an inner automorphism arising from an element of Δ X ). Thus, by Lemma a2.3 below, we conclude that β ◦ α is the identity on X K . Similarly, α ◦ β is the identity on Y K . Lemma a2.3. Let γ : X K → X K be a K-morphism such that π 1 (γ) is the identity on Π X K (up to composition with an inner automorphism arising from an element of Δ X ). Then γ is the identity. 100 Proof. First observe that π 1 (γ) is trivially compatible (up to composition with a geometric . Thus, (the injectivity inner automorphism) with the map induced by π 1 ’s by X K → X K part of) Theorem A tells us that γ is compatible with X K → X K . Let η X be the generic point of X K . Then γ deﬁnes a morphism γ η X : X η X → X η X which induces the identity (up to composition with a geometric inner automorphism) on π 1 ’s. Applying (the injectivity part of) Theorem A again then tells us that γ η X is the identity. Thus, it follows that γ is the identity, as desired. Thus, α and β are isomorphisms. Moreover, it follows from the construction of α that π 1 (α) coincides with φ (up to composition with an inner automorphism arising from an element of Δ Y ), and it follows from Lemma a2.3 that α is the unique such K-isomorphism X K ∼ = Y K . Thus, we see that we see that we have proven the following result: Theorem a2.4. Let K be sub-p-adic (cf. 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